The L(h, k)-Labelling Problem: An Updated Survey and Annotated Bibliography 1
Tiziana Calamoneri
Department of Computer Science “Sapienza” University of Rome - Italy via Salaria 113, 00198 Roma, Italy. Email: [email protected]
Given any fixed nonnegative integer values h and k, the L(h, k)- labelling problem consists in an assignment of nonnegative integers to the nodes of a graph such that adjacent nodes receive values which differ by at least h, and nodes connected by a 2 length path receive values which differ by at least k. The span of an L(h, k)-labelling is the difference between the largest and the smallest assigned frequency. The goal of the problem is to find out an L(h, k)-labelling with minimum span. The L(h, k)-labelling problem has been intensively studied following many approaches and restricted to many special cases, concerning both the values of h and k and the considered classes of graphs. This paper reviews the results from previous by published literature, looking at the problem with a graph algorithmic approach. It is an update of a previous survey written by the same author.
Keywords: L(h, k)-labelling; frequency assignment; radiocoloring; λ-coloring; distance-2-coloring; D2-vertex coloring Received February 2011; revised 00 Month 2011
1. INTRODUCTION follows:
One of the key topics in graph theory is graph Definition 1.1. Given a graph G = coloring. Fascinating generalizations of the no- (V, E) and two nonnegative integers h and tion of graph coloring are motivated by prob- k, an L(h, k)-labelling is an assignment of lems of channel assignment in wireless commu- nonnegative integers to the nodes of G such nications, traffic phasing, fleet maintenance, that adjacent nodes are labelled using colors task assignment, and other applications. (See at least h apart, and nodes having a common [156] for a survey.) neighbour are labelled using colors at least While in the classical vertex coloring k apart. The aim of the L(h, k)-labelling problem [105] a condition is imposed only on problem is to minimize the span σh,k(G), colors of adjacent nodes, many generalizations i.e. the difference between the largest and the require colors to respect stronger conditions, smallest used colors. The minimum span over e.g. restrictions are imposed on colors both of all possible labelling functions is denoted by adjacent nodes and of nodes at distance 2 in λh,k(G) and is called λh,k-number of G. the graph. This paper will focus on a specific graph Observe that this definition imposes a coloring generalization that arose from a condition on labels of nodes connected by a channel assignment problem in radio networks 2 length path instead of using the concept [90]: the L(h, k)-labelling problem, defined as of distance 2, that is very common in the
The Computer Journal, Vol. ??, No. ??, ???? 2 T. Calamoneri literature. The reason is that this definition L(p, q)-coloring problem, distance-2-coloring works both when h k and when h < k. and D2-vertex coloring problem (when h = The present formulation≥ allows the nodes of a k = 1), radiocoloring problem and λ-coloring triangle to be labelled with three colors at least problem (when h = 2 and k = 1). max h, k apart from each other, although Many variants of the problem have been { } they are at mutual distance 1; when h k introduced in the literature, as well: instead of ≥ the two definitions coincide. minimizing the span, seek the L(h, k)-labelling Furthermore, as the smallest used color that minimizes the order, i.e. the number of is usually 0, an L(h, k)-labelling with span effectively used colors [90]; given a span σ, σh,k(G) can use σh,k(G) + 1 different colors; decide whether it is possible to L(h, k)-label this feature is slightly counter-intuitive, but is the input graph using all colors between 0 and kept for historical reasons. σ (no-hole L(h, k)-labelling) [157]; consider the The notion of L(h, k)-labelling was intro- color set as a cyclic interval, i.e. the distance duced by Griggs and Yeh in the special case between two labels i, j 0, 1,...,σ defined h = 2 and k = 1 [89, 184] in connection as min i j , σ + 1 ∈i { j [101];} use a with the problem of assigning frequencies in a more general{| − | model in− which | − |} the labels and multihop radio network (for a survey on the separations are real numbers [86]; generalize class of frequency assignment problems, see the problem to the case when the metric is e.g. [2, 58, 123, 144]), although it has been described by a graph H (H(h, k)-labelling) previously mentioned by Roberts [155] in his [68]; consider the precoloring extension, where summary on T -colorings and investigated in some nodes of the graph are given as already the special case h = 1 and k = 1 as a com- (pre)colored, and the question is if this binatorial problem and hence without any ref- precoloring can be extended to a proper erence to channel assignment (see for instance coloring of the entire graph using a given [180]). number of colors [71]; consider a one-to- After its definition, the L(h, k)-labelling one L(h, k)-labelling (L′(h, k)-labelling) [42]; problem has been used to model several L(h, k)-label a digraph, where the distance problems, for certain values of h and k. from a node x to a node y is the length of Some examples are the following: Bertossi a shortest dipath from x to y [40]; study and Bonuccelli [16] introduced a kind of another parameter, called edge-span, defined integer ”control code” assignment in packet as the minimum, over all feasible labellings, radio networks to avoid hidden collisions, of the max f(u) f(v) : (u, v) E(G) equivalent to the L(0, 1)-labelling problem; [185]; impose{| the labelling− | to be balanced,∈ i.e.} channel assignment in optical cluster based all colors must be used more or less the same networks [10] can be seen either as the L(0, 1)- number of times (equitable coloring) [133]. or as the L(1, 1)-labelling problem, depending Some of these generalizations are considered on the fact that the clusters can contain one in [183]. ore more nodes; more in general, channel assignment problems, with a channel defined The extent of the literature and the huge as a frequency, a time slot, a control code, number of papers concerning the L(h, k)- etc., can be modeled by an L(h, k)-labelling labelling problem have been the main moti- problem, for convenient values of h and k. vation of the surveys [2, 29, 183], each one ap- Besides the practical aspects, also purely proaching the problem from a different point theoretical questions are very interesting. of view (operative research, graph algorithms These are only some reasons why there is and extremal combinatorial, respectively), but considerable literature devoted to the study they are all published at least five years ago. of the L(h, k)-labelling problem, following Since a substantial progress has been achieved many different approaches, including graph in the last years, the author thinks that an theory and combinatorics [156, 169], simulated updated survey and annotated bibliography annealing [54, 141], genetic algorithms [50, would be useful. The present paper is an up- 132], tabu search [39], and neural networks date of [29]. [76, 131]. In all these contexts, the problem In this work, the case k = 0, for any fixed h, has been called with different names; among is not considered as this problem becomes the others, we recall: L(h, k)-labelling problem, classical vertex coloring problem. Instead, a
The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 3 particular accent is posed on the special cases Studying a completely different problem h =0, 1, 2 and k = 1: the first one is equivalent (Hessian matrices of certain non linear func- to the problem of optimally coloring the square tions), McCormick [142] gives a greedy algo- of the input graph and the second one has been rithm that guarantees a O(√n)-approximation considered in the seminal works by Roberts, for coloring the square of a graph. The algo- Griggs and Yeh. Both these problems have rithm is based on the greedy technique: con- been intensively studied. sider the nodes in any order, then the color The decision version of the L(h, k)-labelling assigned to node vi is the smallest color that problem has been proved to be NP-complete, has not been used by any node which is at even under restrictive hypotheses. Section 2 distance at most 2 from vi; the performance lists these results. In Section 3 some general ratio is obtained by simple considerations on the degree of G and of its square. lower and upper bounds on the value of λh,k are summarized. Approaching an equivalent scheduling prob- For some special classes of graphs a labelling lem, Ramanathan and Lloyd [152] present an can be computed efficiently, while for other approximation algorithm with a performance classes of graphs only approximate algorithms guarantee of O(θ), where θ is the thickness of are known. Both these kinds of results are the graph. Intuitively, the thickness of a graph described in Section 4. measures ”its nearness to planarity”. More formally, the thickness of a graph G = (V, E) In the rest of this paper we will consider is the minimum number of subsets into which simple and loopless graphs with n nodes, the edge set E must be partitioned so that each maximum degree ∆, chromatic number χ(G), subset in the partition forms a planar graph on clique number ω(G) and girth (i.e. the V . length of the shortest cycle in G) g(G). For all graph theoretic concepts, definitions and L(2, 1)-labelling. To decide whether a graph classes inclusions not given in this given graph G admits an L(2, 1)-labelling of review we refer either to [26] or to the related span at most n is NP-complete [89]. This reference. result is obtained by a double reduction: from HAMILTONIAN PATH to the decision 2. NP-COMPLETENESS RESULTS problem asking for the existence of an injection f : V [0,n 1] such that f(x) f(y) 2 In this section some general complexity results whenever→ (x, y−) E, and from| this− problem|≥ are listed, divided by different values of h and to the decisonal version∈ of the L(2, 1)-labelling k. More specific results concerning classes of problem. The problem remains NP-complete graphs are given in Section 4. if we ask whether there exists a labelling of L(0, 1)-labelling. In [16] the NP- span at most σ, where σ is a fixed constant completeness result for the decision version of 4, while it is polynomial if σ 3 (this case ≥ ≤ the L(0, 1)-labelling problem is derived when occurs only when G is a disjoint union of paths the graph is planar by means of a reduction of length at most 3). A fortiori, the problem from 3-VERTEX COLORING of straight-line is not fixed parameter tractable [67]. planar graphs. The problems of finding the λ2,1-number of graphs with diameter 2 [89, 184], planar L(1, 1)-labelling. Also the decision version graphs [20, 57], bipartite, split and chordal of the L(1, 1)-labelling problem, (that is graphs [20] are all NP-hard. equivalent to the L(2, 1)-labelling problem Finally, Fiala and Kratochv´ıl[69] prove that where the order must be minimized instead for every integer p 3 it is NP-complete to of the span [74]) is proved to be NP-complete decide whether a p-regular≥ graph admits an with a reduction from 3-SAT [142]. The L(2, 1)-labelling of span (at most) p + 2. problem remains NP-complete for unit disk graphs [161], for planar graphs [151] and even We conclude this paragraph citing some for planar graphs of bounded degree [150]. It results where the authors present exact is also NP-complete to decide whether 4 colors exponential time algorithms for the L(2, 1)- suffice to L(1, 1)-label a cubic graph. On the labelling problem of fixed span σ. In contrary, it is polynomial to decide if 3 colors [128], the authors design algorithms that are are enough [99]. faster than the naive O((σ + 1)n) algorithm
The Computer Journal, Vol. ??, No. ??, ???? 4 T. Calamoneri that would try all possible labellings. In h + 4 as the last open cases for the the first NP-complete case (σ = 4), the fixed parameter complexity of the L(h, 1)- running time of their algorithm is O(1.3006n), labelling problem. which beats not only O(σn) but also O((σ n − 1) ). For what concerns the larger values 3. LOWER AND UPPER BOUNDS of σ, an exact algorithm for the so called Channel Assignment Problem [125] implies an We list here some general bounds on the λh,k- a O(4n) algorithm for the L(2, 1)-labelling number, divided by different values of h and problem. This has been improved in [96] to k. Bounds for particular classes of graphs will an O(3.8739n) algorithm. Some modifications be given in the corresponding subsections. of the algorithm and a refinement of the L(1, 1)-labelling. Define f(∆,g) as the running time analysis has allowed to improve 2 n maximum possible value of λ1,1(G)= χ(G ) the time complexity to O(3.2361 ) [115]. A 1 over graphs with maximum degree ∆ and− lower-bound of Ω(3.0731n) on the worst-case girth g. Since the maximum degree of G2 is running time is also provided. After this at most ∆2, it follows that f(∆,g) ∆2 for result, the base of the exponential running every g. This bound is tight as equality≤ holds time function seemed hardly decreaseable to for ∆ = 2 and g 5, as shown by the 5-cycle, a value lower than 3. Nevertheless, in [116] for ∆ = 3 and g ≤5, as shown by the Petersen the authors provide a breakthrough in this graph, and for ∆≤ = 7 and g 5, as shown by question by providing an algorithm running ≤ n the Hoffman-Singleton graph. (The Hoffman- in O(2.6488 ) time, based on a reduction of Singleton graph – see Figure 1 – is the graph the number of operations performed in the on 50 nodes and 175 edges that is the only recursive step of the dynamic programming regular graph of node degree 7, diameter 2, algorithm. and girth 5; it is the unique (7, 5)-cage graph L(h, k)-labelling. Nobody would expect and Moore graph, and contains many copies the L(h, k)-labelling problem for h > k 1 of the Petersen graph). Moreover, by Brooks to be easier than the L(2, 1)-labelling problem,≥ theorem (stating that if G is connected then however, the actual NP-hardness proofs seem χ(G) ∆(G), unless G is complete or G is ≤ tedious and not easily achievable in full an odd cycle; cf. e.g. [105]) it follows that generality. In [67] the authors conjecture that the equality can hold only for g 5 and only ≤ for every h k 1, there is a σ (depending if there exists a ∆-regular graph of diameter on h and ≥k) such≥ that deciding whether 2 on ∆2 + 1 nodes. If such graph exists then λ (G) σ is NP-complete. In support of ∆ 2, 3, 7, 57 . It is also possible to see that h,k ∈{ } their conjecture,≤ the authors prove that there f(2,g) = 4 for all g 6. Alon and Mohar [9] ≥ is at least one NP-complete fixed parameter prove that f(∆,g) is (1 + o(1))∆2 if g =3, 4, 5 instance, namely that it is NP-complete to and is Θ(∆2/ log ∆) if g 7. In [145] a new ≥ decide whether λ (G) h+k h for all fixed approach is followed in order to provide a new h,k ≤ ⌈ k ⌉ h > k 1. Under less general conditions they upper bound on λ1,1. Namely, utilizing the prove that≥ there are infinitely many instances: probabilistic method, the authors prove that 2 2 the problem whether λ (G) h + pk is NP- f(∆,g) 1 6 ∆ if the graph is regular h,k ≤ ≤ − 3e complete for any fixed p h and h > 2k. It of girth g 7 and ∆ sufficiently large. k ≥ follows that for k = 1 (and≥ more generally h L(2, 1)-labelling. As in the case of λ , divisible by k), there are only finitely many 1,1 even for the bounds on λ , ∆ is the most polynomial instances (unless P=NP), namely 2,1 common used parameter. if h > 2 then the decision version of the The obvious lower bound for λ (G) is L(h, 1)-labelling problem is NP-complete for 2,1 ∆ + 1, achieved for the star K , but Griggs every fixed σ 2h. In this case it is possible 1,∆ and Yeh [89] describe a graph requiring span a little more:≥ for every h > 2, the problem of ∆2 ∆. This graph is the incidence graph deciding whether λ (G) σ is NP-complete h,1 of a− projective plane π(n) of order n, i.e. the if σ h + 5 while it is polynomially≤ solvable if ≥ bipartite graph G = (U V, E) such that: σ h + 2. ∪ ≤ i. U = V = n2 + n + 1, | | | | Open Problem: For p 5 this result ii. each u U corresponds to a point pu in leaves the cases σ = ≥h + 3 and σ = π(n) and each∈ v V corresponds to a line l ∈ v The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 5
4 0 4 0 4 0 4 0 4 0
P P PPP 3 0 1 3 1 1 3 21 3 31 3 4 1
2 2 2 2 2
1 1 1 1 1
2 0 2 0 2 0 2 0 2 0
QQQQQ0 1 2 3 4
3 4 3 4 3 4 3 4 3 4
FIGURE 1. The Hoffman-Singleton graph, constructed from the 10 5-cycles illustrated, with node i of Pj joined to node i + jk(mod 5) of Qk.
in π(n), and an open problem to understand if there iii. E = (u, v): u U, v V such that are infinitely many graphs G satisfying { ∈ ∈ 2 pu lv in π(n) . λ2,1(G) > ∆ o(∆). By∈ definition of} π(n), G is (n + 1)-regular. − By a simple greedy algorithm, that labels Using constructive labelling schemes, Jonas each node with the smallest color that does not [112] improves the upper bound by showing 2 induce conflicts, the same authors prove that that λ2,1(G) ∆ + 2∆ 4 if ∆ 2 2 ≤ − ≥ λ2,1(G) ∆ + 2∆ and improve this upper and, successively, Chang and Kuo [42] further ≤ 2 bound to λ2,1(G) ∆ + 2∆ 3 when G is decrease the bound to ∆2 + ∆. The algorithm ≤ 2− 3-connected and λ2,1(G) ∆ when G is of by Chang and Kuo is funded on the concept of ≤ 2 diameter 2. They conjecture λ2,1(G) ∆ 2-stable set of a graph G, that is a subset S of ≤ for any graph G. This conjecture has been V (G) such that every two distinct nodes in S motivation of some research since. In fact, are of distance greater than 2. At each step i we can claim that this is the most famous of the algorithm, a subset of nodes Si is built, open problem in this area for more than and all nodes of Si are labelled with i. Si is fifteen years. In his invited talk, during the a maximal 2-stable set of the set of unlabelled conference CIAC 2010, Reed has claimed that nodes at distance 2 from any node in Si 1. ≥ − the Griggs and Yeh’s conjecture has raised an Then, it has been proven the analogue of interest that can be compared with the one Brook’s theorem for some channel assignment given to the Four Colors’ conjecture. problems, deriving as corollary of a more 2 Observe that the upper bound set by the general result that λ2,1(G) ∆ + ∆ 1 for conjecture would be tight: there are graphs any graph G [127], and successively≤ Gon¸calves− 2 2 with degree ∆, diameter 2 and ∆ + 1 nodes, [84] proved λ2,1(G) ∆ + ∆ 2 using the namely the 5-cycle, the Petersen graph and the algorithm of Chang and≤ Kuo. In− 2008, Havet, Hoffman-Singleton graph, so the span of every Reed and Sereni [97] have finally proven that L(2, 1)-labelling is at least ∆2. Nevertheless, the Griggs and Yeh’s conjecture is true, for notice that the conjecture is not true for ∆ = sufficiently large values of ∆ (about ∆ 1069). ≥ 1. For example, ∆(K2) = 1 but λ2,1(K2) = 2. This is the first paper addressing the solution of the conjecture for general graphs, although Open Problem: The Moore graphs (i.e. it does not completely close it. The same graphs having the minimum number of authors prove in [98] that for every graph G 2 nodes possible for a regular graph with of maximum degree ∆, λ2,1(G) ∆ + c, given diameter and maximum degree) for some constant c. Of course, the≤ bound of are at the moment the only graphs ∆2 +c is better than ∆2 +∆ 1 as ∆ increases known to require span ∆2, and it is enough, but c is unfortunately− a rather huge
The Computer Journal, Vol. ??, No. ??, ???? 6 T. Calamoneri number. The interest of this result lies in degree ∆. This bound has been improved to 2 having exactly determined the growth of the λh,1(G) ∆ + (h 1)∆ 2 when ∆ 3 2 ≤ − −λh+1,1(G) ≥ function to be added to ∆ . Now, it is a [84]. Furthermore, lim h = 1 [41]. →∞ λh,1(G) very interesting issue to find a tight value for Havet, Reed and Sereni [97] generalize their c (hopefully as close to 0 as possible). proof of the Griggs and Yeh’s conjecture for Open Problem: The Griggs and Yeh’s sufficiently large values of ∆ to any h and k, 2 69 conjecture is still unproved for ∆ < 1069 proving that λh,1(G) ∆ for any ∆ 10 2 ≤ ≥ and hence it is not considered closed yet. and λh,1 ∆ + c(h) for every integer ∆ and for an opportune≤ constant c(h), depending on For what concerns the relationship between the parameter h. λ2,1 and the graph parameters, it is not It is easy to see that λ (G) h + (∆ 1)k h,k ≥ − difficult to see that λ2,1 2ω(G) 2. for h k. Moreover, if h > k and the equality ≥ − ≥ A relationship between λ2,1 and the chro- holds in the previous formula and h > k, then matic number of G is stated in [89]: λ2,1(G) for any L(h, k)-labelling of G, each node of ≤ n+χ(G) 2 and for complete k-partite graphs degree ∆ must be labeled 0 (or h + (∆ 1)k) − − the equality holds, i.e. λ2,1(G)= n + k 2. and its neighbors must be labeled h + ik (or − In [83] the authors investigate the relation- ik) for i =0, 1,... ∆ 1. − ship between λ2,1(G) and another graph in- The structures of graphs with ∆ 1 and ≥ variant, i.e. the path covering number c of λh,k(G)= h + (∆ 1)k are studied in [43] and C − the complement graph G (the path cover- they are called λh,k-minimal graphs. ing number of a graph is the smallest num- ber of node-disjoint paths needed to cover the A basic result, implicitly taken into account in any work on the L(h, k)-labelling, states nodes of the graph) proving that λ2,1(G) = n + c(GC ) 2 if and only if c(GC ) 1. that for all G, there exists an optimal L(h, k)- − ≥ labelling of G such that each label is of the It is quite simple to see that λ2,1 and λ1,1 are related: the number of colors necessary form αh + βk, α and β being non negative for an L(2, 1)-labelling of a graph G is at integers. Hence, in particular, λh,k(G) = αh + βk for some non negative integers α and least λ1,1 + 1, and conversely from an optimal L(1, 1)-labelling of G we can easily obtain an β [78]. It is also worthy to notice that, for any positive integer c, c λ (G) = λ (G) L(2, 1)-labelling of G with colors between 0 · h,k ch,ck and that if h′ h and k′ k, then for any and 2λ1,1 1. Hence, an algorithm solving ≥ ≥ − graph G, λ ′ ′ (G) λ (G) [78]. Finally, let the L(1, 1)-labelling problem for a class of h ,k ≥ h,k graphs also provides a 2-approximation for the G have maximum degree ∆. Suppose there is L(2, 1)-labelling problem. a node with ∆ neighbors, each of which has degree ∆. Then λ (G) h + (2∆ 2)k if In [13], Balakrishnan and Deo give upper h,k ≥ − and lower bounds on the sum and product of h ∆k and λh,k(G) 2h+(∆ 2)k if h ∆k. Of≥ course, these lower≥ bounds− fit particularly≤ the λ2,1-number of an n node graph and that of its complement: well for regular graphs. In [38], by stating a strong relationship C 2√n 2 λ2,1(G)+ λ2,1(G ) 3n 3 between the L(h, k)-labelling problem and − ≤ ≤ − the problem of coloring the square of a 3n 3 2 graph, it is exploited the algorithm by 0 λ (G) λ (GC ) − ≤ 2,1 · 2,1 ≤ 2 McCormick for approximating the L(1, 1)- labelling problem [142] to provide a (h√n + These bounds are similar to the well consoli- o(h√n))-approximation algorithm for the dated bounds given by Nordhaus and Gaddum L(h, k)-labelling problem. This result has [146] on the chromatic number of a graph and been improved by Halld´orson [92] proving that of its complement. that the performance ratio of the First Fit L(h, k)-labelling. Passing to the general algorithm (consisting in processing the nodes L(h, k)-labelling problem, for any positive in an arbitrary order, so that each node is integers h k, λh,k h + (∆ + 1)k [43], assigned the smallest color compatible with its ≥ ≥ h as a generalization of the known results for neighborhood) is at most O(min(∆, k + √n)). h = 2, it is easy to state that λh,1(G) This is tight within a constant factor, for all ∆2 + (h 1)∆ for any graph of maximum≤ values of the parameters. The author shows − The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 7
that this is close to the best possible, as it is cycles, having λ0,1(Cn) equal to 1 if n is NP-hard to approximate the L(h, k)-labelling multiple of 4, and 2 otherwise [16]. In Figure 1/2 ǫ problem within a factor of n − for any ǫ> 0 2 these labellings are shown. 1/2 ǫ and h in the range [n − ,n]. On the positive It is easy to check that λ0,1(Kn) = side, it is never harder to approximate than λ (K ) = n 1 and that λ (W ) = 1,1 n − 0,1 n the ordinary vertex coloring problem, hence an λ1,1(Wn) = n according to our definition of 2 3 upper bound of O(n(log log n) / log n) holds L(h, k)-labelling; on the contrary, λ0,1(Wn)= [91]. (n 1) −2 according to the definition based on the⌊ concept⌋ of distance. 4. KNOWN RESULTS ON GRAPH Furthermore, λ1,1(P2) = 1 and λ1,1(Pn)=2 CLASSES for each n 3; λ1,1(Cn) is 2 if n is a multiple of 3 and it≥ is 3 otherwise [14] (see Figure 3). In view of the hardness results described in Section 2 and of the gap between upper and L(2, 1)-labelling. It is simple to prove lower bounds on the λh,k-number listed in that λ2,1(P1) = 0, λ2,1(P2) = 2, λ2,1(P3) = Section 3, further bounds, exact results and λ2,1(P4) = 3, and λ2,1(Pn) = 4 for n 5, ≥ approximation algorithms have been found that λ2,1(Kn) = 2(n 1), that λ2,1(W3) = − by restricting the classes of graphs under λ2,1(W4) = 6 and λ2,1(Wn) = n + 1 for each consideration. This is the topic of the present n 5. Finally, λ2,1(Cn) = 4 for each n 3 ≥ ≥ section. [89, 184]. The L(h, k)-labelling problem has been As an example, we recall here how to label intensively studied on various graph classes a cycle. If n 4 the result is trivial to ≤ in its general version (any h and k) but verify. Thus, suppose that n 5, and Cn must ≥ above all in some of its specializations (e.g. contain a P5 as a subgraph, hence λ2,1(Cn) ≥ h = 2, 1 and k = 1); some of these classes λ2,1(P5) = 4. Now, let us show an L(2, 1)- have been considered because they well model labelling l of Cn with span 4. Let v0,...,vn 1 − real networks, others for their theoretical be nodes of Cn such that vi is adjacent to vi+1, interest. In the following we analyze a number 0 i n 2 and v0 is adjacent to vn 1. Then, ≤ ≤ − − of graph classes, describe the known results consider the following labelling: concerning each of them, and propose some if n 0 mod 3 then ≡ interesting problems still open. This section 0 if i 0 mod 3 is organized listing first the graph classes ≡ l(vi)= 2 if i 1 mod 3 for which exact results are known, and then 4 if i ≡ 2 mod 3 the graph classes for which only approximate ≡ bounds and labeling algorithms have been if n 1 mod 3 then redefine l at ≡ found. In view of this organization, not all vn 4,...,vn 1 as − − correlated classes are treated in consecutive 0 if i n 4 subsections. ≡ − 3 if i n 3 l(vi)= ≡ − 1 if i n 2 4.1. Paths, Cycles, Cliques and Wheels 4 if i ≡ n − 1 ≡ − Let P , C and K be a path, a cycle and n n n if n 2 mod 3 then redefine l at vn 2 and a clique, respectively, of n nodes. The wheel ≡ − at vn 1 as − Wn is obtained by Cn by adding a new node adjacent to all nodes in Cn. 1 if i = n 2 l(vi)= − Paths (i.e. buses), cycles (i.e. rings), 3 if i = n 1 − cliques (i.e. completely connected networks) Polynomial results concerning L(2, 1)-labelling and wheels are the simplest and most common are found also for classes of cycle-related networks one can consider; the decision graphs such as cacti, unicycles and bicycles version of the L(h, k)-labelling problem is [112]. (A cactus is a connected finite graph polynomially solvable on each of them. in which every edge is contained in at most L(0, 1)- and L(1, 1)-labelling. Optimal one cycle; a unicycle – respectively, bicycle L(0, 1)-labellings are known for paths, needing – is a connected graph having only one – 2 colors (i.e. λ0,1(Pn) = 1) [139], and for respectively, two – cycles.)
The Computer Journal, Vol. ??, No. ??, ???? 8 T. Calamoneri
0 0 1 1 0 0 0 1 1
0 0 1 1 0 0 1
1 0 2 0
1 0 0 1 1 2 1 1 0
a. b. c.
FIGURE 2. a. L(0, 1)-labelling of a path; b. L(0, 1)-labelling of a cycle whose number of nodes is multiple of 4; c. L(0, 1)-labelling of a cycle whose number of nodes is not multiple of 4.
0 1 2 0 1 0 1 2 0 0 1 2 0 0 1
0 1 2 0 1 2 0 2 2 3 1
1 0 2 1 0 2 1 0 2 3 2 1 3
a. b. c. d.
FIGURE 3. a. L(1, 1)-labelling of a path; b. L(1, 1)-labelling of a cycle whose number of nodes is multiple of 3; c. and d. L(1, 1)-labellings of cycles whose numbers of nodes are not multiple of 3.
L(h, k)-labelling. Georges and Mauro [78] tively. Portions of these grids are shown in evaluate the span of cycles and paths for any Figure 4. h and k (with h k) showing that The hexagonal grid is a natural model for ≥ cellular networks, and its interference graphs 0 if n =1 is the triangular grid, also called cellular graph, h if n =2 according to the notation introduced in [163]. λ (P )= h + k if n =3or4 The L(h, k)-labelling problem has been h,k n h +2k if n 5 and h 2k extensively studied on regular grids, and 2h if n ≥ 5 and h ≥ 2k shown to be polynomially solvable. More ≥ ≤ detailed results are given in the following. and Note that some grids are equivalent to some special products of paths, so other related 2h if n odd, n 3 and h 2k,results can be found in Subsection 4.3. ≥ ≥ or if n 0 mod 3 and h 2k L(0, 1)-, L(1, 1)- and L(2, 1)-labelling. ≡ ≤ h +2k if n = 0 mod 4 and h 2k,An optimal L(0, 1)-labelling for squared grids ≥ or if n 0 mod 3, n = 5 andandh an2 optimalk L(1, 1)-labelling for hexagonal, λh,k(Cn)= 6≡ 6 ≤ 2h if n = 2 mod 4 and h 3k squared and triangular grids are given in [139] ≤ h +3k if n = 2 mod 4 and h 3k and in [14], respectively. ≥ 2h if n 5 and h 2k The λ -number of regular grids has been ≥ ≤ 2,1 4k if n =5 proved to be λ (G ) = ∆+2 by means 2,1 ∆ of optimal labelling algorithms [35] . All of It is straightforward to see that an optimal these algoithms are based on the replication L(h, k)-labelling of an n node clique requires of a labelling pattern, depending on ∆. An span (n 1)h, for each h k and that example is given in Figure 5. λ (W )− = n + h 1 for sufficiently≥ large h,k n A natural generalization of squared grids is values of n and h k−. obtained by adding wrap-around edges on each Finally, we point≥ out that the λ -number h,1 row and column: these graphs are known as of cacti is investigated in [121]. tori. In spite of the similarity between squared grids and tori, the presence of wrap-around 4.2. Regular Grids edges prevents the labelling of the squared grid Let G∆,∆=3, 4, 6, 8, denote the hexagonal, from being extended to tori unless both the squared, triangular and octagonal grid, respec- number of rows and the number of columns
The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 9
a. b. c. d.
FIGURE 4. A portion of: a. hexagonal grid G3; b. squared grid G4; c. triangular grid G6; d. octagonal grid G8.
1 3
1 3 1 3 3 5 0 2 7 3 5 0 2 3 5 0 2
f2 f 6 f2 2 4 f1 0 2 4 1 6 0 f2 2 4 f1 1 5 7 8 f3 4 0 f3 f3 4 6 1 3 1 3 4 6 1 3 8 3 5 3 5
3 5
a. b. c.
FIGURE 5. An example of L(2, 1)-labelling of regular grids by means of labelling patterns; function f1 subtracts 1 to each label, f2 sums 2 and f3 sums 1; all the operations must be considered mod (λ2,1(G∆) + 1).
are multiples of 5. If this is not the case, an obtained for any h and k include as special L(1, 1)-labelling exists using at most 8 colors, case the previous ones for h =1, 2 and k = 1. which is nearly-optimal since 6 is a lower All the aforementioned results lead to assign bound [52]. Exact results concerning the λ2,1- a color to any node in constant time in a number of tori are reported in Subsection 4.3. distributed fashion, provided that the relative positions of the nodes in the grid are locally Open Problem: To find the optimal L(1, 1)- known. labelling of tori is an interesting issue. Later, Griggs and Jin [87] close all gaps for the squared and hexagonal grids, and all gaps L(h, k)-labelling. In 1995 Georges and except when k/2 h 4k/5 for the triangular Mauro [78] give some results concerning the grids; they do not≤ handle≤ the octagonal grid. L(h, k)-labelling of squared grids as a special A summary of all the known results is plotted result of their investigation on the λ - h,k in Figure 6. number of product of paths. Only in 2006 the problem has been systematically handled Open Problem: It remains to compute the and the union of the results presented in [30] exact value of λh,k(G∆) in those intervals and in [31] provides the exact value of function where lower and upper bounds do not λh,k(G∆), where ∆ = 3, 4, 6, 8 for almost all coincide when ∆ = 6 and ∆ = 8. values of h and k. Open Problem: Almost all the proofs for The exact results are obtained by means the lower bounds are based on exhaustive of two series of proofs: lower bounds proofs, reasonings, and so are very long and based on exhaustive considerations, deducing difficult to follow. Furthermore, the range that λh,k(G∆) cannot be less than certain of h/k is divided into several intervals, values, and upper bounds proofs, based on and a different proof is given for each labelling schemes. Of course, the results ∆ and for each interval. It would
The Computer Journal, Vol. ??, No. ??, ???? 10 T. Calamoneri
λ"
#&!" #%!" #$!" .)" ##!" #+!" #,!" +*!" +)!" +(!" .'" +'!" +&!" +%!" +%!" +$!" +#!" .%" ++!" +,!" +,!" ++!" .$" *!" )!" )!" )!" )!" (!" '!" '!" '!" &!" %!" &!" %!" $!" #!" $!" !"
!" #!" $!" %!" &!" '!" -"
FIGURE 6. State of the art concerning the L(h, k)-labelling of regular grids. Grey areas represent the gaps to be still closed.
be interesting to design a new proof technical, without any particular practi- technique in order to simplify all these cal relevance. proofs and to propose a unifying approach useful to reduce the number of the proofs 4.3. Product of Paths, Cycles and and to increase their elegance. Cliques
A generalization of the squared grid is The cartesian product (or simply product) G2H and the direct product G H of the d-dimensional grid. A motivation for × studying higher dimensional grids is that graphs G and H are defined as follows: 2 when the networks of several service providers V (G H)= V (G H) is equal to the cartesian × 2 overlap geographically, they must use different product of V (G) and V (H); E(G H) = channels for their clients. The overall network ((x1, x2), (y1,y2)) : (x1,y1) E(G) and x{ = y , or (x ,y ) E(H) and∈ x = y ; can then be modelled in a suitably higher 2 2 2 2 ∈ 1 1} dimension. Optimal L(2, 1)-labellings for d- E(G H) = ((x1, x2), (y1,y2)) : (x1,y1) × { 2 ∈ dimensional square grids, for each d 1 are E(G) and (x2,y2) E(H) . G H and ≥ G H are mutually∈ nonisomorphic} with the presented in [53]. These results are extended × to any h, k for each d 1 in [61]. The authors sole exception of when G and H are odd cycles ≥ of the same size. The strong product G H of give lower and upper bounds on λh,k for d- ⊗ dimensional grids, and show that in some cases G and H has the same node set as the other these bounds coincide. In particular, in the two products and the edge set is the union of E(G2H) and E(G H). In Figure 7 the case k = 1, the results are optimal. × product of P3 and P4 is depicted, according to Open Problem: It is still an open ques- each one of the three just defined products. tion to find optimal, or nearly-optimal, labellings for higher dimensional triangu- Product graphs have been considered in the lar grids, even for special values of h and attempt of gaining global information from the k. Nevertheless, it is in the opinion of the factors. Many interesting wireless networks author that such a result would be only have simple factors, such as paths and cycles.
The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 11
! x ⊗
a. b. c.
FIGURE 7. Product of P3 and P4, where the product is: a. the cartesian product; b. the direct product and c. the strong product.
Observe that any d-dimensional grid is with m = 3 or m multiple of 4 or 5. the cartesian product of d paths, any d- Finally, in [160] the previous partial results on dimensional torus is the cartesian product λ2,1(Cm2Cn) are completed; for all values of of d cycles and any octagonal grid is the m and n, λ2,1(Cm2Cn) is either 6 or 7 or 8, strong product of two paths; so there according to the values of m and n. is some intersection between the results Exact results for the L(2, 1)-labelling of the summarized in this section and in the product of complete graphs Km2Kn and of previous one. Nevertheless, they have been Kp2 ... 2Kp repeated q times, when p is prime described separately in order to highlight the are given in [82]. independence of the approaches and of the Concerning the direct and strong products, methods for achieving the results. The same Jha obtains the λ2,1-number of some infinite reasoning holds for the cartesian product of families of products of several cycles [106, 107]. complete graphs, i.e. the Hamming graph and Exact values for C C , C C and 3 ⊗ n 4 ⊗ n for the cartesian product of n K2 graphs, i.e. improved bounds for Cn Cm are presented the n-dimensional hypercube: we will detail in [122]. ⊗ the results on these graphs in Subsection 4.8. In [110], the λ2,1-number is computed for C C for some special values of m and n. m × n L(2, 1)-labelling. Exact values for the λ2,1- Papers [117] and [166] handle the L(2, 1)- numbers for the cartesian product of two paths labelling of graphs that are the direct/strong for all values of m and n are given in [182]. product and the cartesian product of general 2 Namely, the authors prove that λ2,1(Pm Pn) non trivial graphs. The authors prove that for is equal to 5 if n = 2 and m 4 and it is equal all the three classes of graphs the conjecture by ≥ to 6 if n,m 4 or if n 3 and m 5. Griggs and Yeh is true. In [164] the previous ≥ ≥ ≥ In the same paper the L(2, 1)-labelling of upper bounds for direct and strong product the cartesian product of several paths P = of graphs are improved. The main tool for n i=1 Ppi is also considered. For certain values this purpose is a more refined analysis of Qof pi exact values of λ2,1(P ) are obtained. neighborhoods in product graphs. Finally, in From this result, they derive an upper bound [165], the authors correct a mistake in a proof for the span of the hypercube Qn. of [166], and study another product similar to In [108] the λ2,1-numbers for the product the cartesian product called composition. of paths and cycles are studied: bounds for λ2,1(Cm2Pn) and λ2,1(Cm2Cn) are given, L(h, k)-labelling. In [109] exact values for and they are exact results for some special the λh,1-number of Cc1 ... Ccn and of 2 2 × × values of m and n. The authors of [118] Cc1 ... Ccn are provided, if there are certain and [130] independently achieve the same conditions on h and on the length of the cycles issue of completing the previous results, and c1,...,cn. determine λ2,1(Cm2Pn) for all values of m and In [77, 82] the L(h, k)-labelling problem of n: λ2,1(Cm2Pn) is either 5 or 6 or 7, according products of complete graphs is considered (h to the values of m and n. k) and the following exact results are given,≥ Kuo and Yan [130] determine λ (C 2C ) where 2 n λ (K 2K ) = (m 1)h + (n 1)k if h >n; child and its right child. Then, consider h,k n m − − k λ (K 2K ) = (mn 1)k if h n; the nodes by increasing levels: if a node h,k n m − k ≤ λh,k(Kn2Kn) = (n 1)h + (2n 1)k if has been assigned label c, then assign the h >n− 1; − remaining two colors to its grandchildren, k − λ (K 2K ) = (n2 1)h if h n 1. but giving different to brother grandchildren. h,k n n − k ≤ − In [60] the λh,k-number of the cartesian The above procedure can be generalized to n find an optimum L(1, 1)-labelling for complete product i=1 Kti is exactly determined for (∆ 1)-ary trees, requiring span ∆. It is n 3 andQ relatively prime t1,...,tn, where − 2 ≥t The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 13 on trees can be also solved in linear time if and k > 1 could be put into relationship ∆=Ω(√n). It follows that the worst running with the difference between systems of distinct time of the algorithm in [94] is O(n1.75). and distant representative [72]. Another Hasunama, Ishii, Ono and Uno have presented result going toward the same direction states a linear time algorithm for L(2, 1)-labelling of that the decisional version of the L(h, k)- trees [94], which finally settles the problem of labelling problem is NP-complete for trees if improving the complexity to linear time and h is part of the input and k 2 is fixed closes the question. [65]. The resolutive step remains≥ to study A tree T is of type 1 if λ2,1(T ) = ∆+1 the computational complexity of the problem and of type 2 if λ2,1(T ) = ∆+2. It seems when both h and k are fixed. that characterizing all type 1 (2) trees is very The paper [64] definitively resolves this difficult. In [175] a sufficient condition for a question proving that for positive integers tree T to be of type 1 is given. Namely, the h and k, the L(h, k)-labelling problem author proves that if a tree T contains no two restricted to trees is solvable in polynomial nodes of maximum degree at distance either 1, time only if k divides h, otherwise it 2 or 4, then λ2,1(T ) = ∆ + 1. is NP-complete. In particular, in the first case the L(h, k)-labelling problem is L(h, k)-labelling. Chang and Kuo’s equivalent to the L(h/k, 1)-labelling problem, algorithm can be generalized to trees and to and hence is solvable in polynomial time by p-almost trees to polynomially determine the the modification of the Chang and Kuo’s exact λh,1-number [41, 67]. algorithm presented in [41]. In the case of For any tree T of maximum degree ∆, ∆+ mutually prime h and k, the NP-hardness is h 1 λ (T ) min ∆+2h 2, 2∆+h 2 proved by a reduction from the problem of − ≤ h,1 ≤ { − − } [41]; the lower and the upper bounds are deciding the existence of a system of distant both attainable. Lower bounds on the λh,1- representatives in systems of symmetric sets. number can be given also as a function of other The main idea is a construction of trees that parameters of the tree (the big-degree and the allow only specific labels on their roots; the neighbour-degree) [121]. main difficulty is to keep the size of such trees While the generalization of this algorithm polynomial. is quite easy when k = 1, the case k > Georges and Mauro provide bounds on the 1 has kept resisting all attempts up to λh,k-number for general h and k for trees when Fiala, Golovach and Kratochvil [64] of maximum degree ∆ h/k [78] and solved the problem, as highlighted below. then for trees with h ≤k and ∆ 3 Before this paper, well known researchers had [80]. For these parameters≥ they obtain≥ tight conjectured both the polynomiality and the upper and lower bounds on λ for infinite NP-hardness of the L(h, k)-labelling problem h,k trees. In [34], the authors present results on trees. Namely, from the one hand Welsh that are complementary, investigating L(h, k)- [181] suggested that, by an algorithm similar labellings of trees, for arbitrary positive to Chang and Kuo’s, it should have been integers h < k, seeking such labellings with possible to determine λh,k(T ) for a tree T small span. The relatively large values of and for arbitrary h and k, hence conjecturing λh,k(T ) achieved are witnessed by trees of that the general case is also polynomial for large height. This fact is not accidental: for trees. From the other hand, based on some trees of height 1, i.e., for stars, the span of considerations concerning the crucial step of L(h, k)-labellings is in fact smaller. Chang and Kuo’s algorithm, Fiala, Kratochv´ıl Finally, an upper bound on λ is given and Proskurowski conjecture that determining h,k in terms of a new parameter for trees, the λ (T ) is NP-hard for trees, when k > 1 h,k maximum ordering-degree [113]. [71]. The feeling that k > 1 identified a more difficult problem seemed to be justified from 4.5. Bounded Width Graphs the fact the problem becomes NP-complete if some nodes of the input tree are precolored, 4.5.1. Bounded Clique-Width Graphs whereas for k = 1 the precolored version The clique-width of a graph G, denoted by remains anyway polynomially solvable [71]. cwd(G), is the minimum number of labels Furthermore, the difference between k = 1 needed to construct graph G using four The Computer Journal, Vol. ??, No. ??, ???? 14 T. Calamoneri operators: , +, ρ and η. The operation i the graph whose node set is V (G) V (H) and creates a graph· with a single node labelled i·. edge set is E(G) E(H). The join ∪of G and H, The binary operator + constructs the union G + H, is the graph∪ obtained from G H by ∪ of two disjoint graphs. The operation ρi j adding all edges between nodes in V (G) and renames all nodes labelled i with label j. The→ nodes in V (H). unary operator ηi,j , with i = j, adds all the Cographs are defined recursively by the edges between every node labelled6 i and every following rules: node labeled j. The sequence of operations 1. A node is a cograph; produces an expression of the graph. 2. if G and H are cographs, then so is their Many NP-hard problems become polyno- join G + H; mially solvable for graphs of bounded clique- 3. if G and H are cographs, then so is their width, if an expression of the graph is part of union G H. ∪ the input; the classical vertex coloring prob- Chang and Kuo [42], as a consequence lem is one of them. Observe also that sev- of their more general result concerning the eral classes of graphs (partial t-trees – that L(2, 1)-labelling problem on union and join will be extensively treated in Subsection 4.5.2, of graphs and exploiting the linear time distance-hereditary graphs, P4-sparse graphs, algorithm to identify whether a graph is a P4-tidy graphs, etc.) are known to have cograph [46], prove that there is a linear time bounded clique-width. algorithm to compute λ2,1(G) for a cograph G. L(1, 1)-labelling. The problem is polyno- Open Problem: Of course, the polynomi- mial when restricted to graphs with bounded ality of the L(2, 1)-labelling problem on clique-width. Indeed, for any graph G of cographs does not implies anything for 2 clique-width t, the clique-width of G is at the L(h, k)-labelling problem. In fact, it most t 2t+1. So, an approach consists in us- · is still unknown the computational com- ing the vertex coloring algorithm for graphs plexity of the L(h, k)-labelling problem on of bounded clique-width proposed in [119] on cographs, and no algorithms are known. G2. Suchan and Todinca [170] propose an al- ternative algorithm, that improves the com- 4·2t log 3+1 L(h, k)-labelling. It is known [48] putational complexity from O(n2 ) to that all problems expressible in MS1-logic 3 2t log 3 4 O(n · n ). Although the complexity re- are fixed parameter tractable (FPT), when mains high, it is considerably lower than the parameterized by the clique-width of the input previous one. graph. Hence deciding whether λh,k(G) σ for a fixed value of σ is polynomial for graphs≤ L(2, 1)-labelling. The decisional version of of bounded clique-width. the problem is NP-complete even for graphs of clique-width at most 3 [21]. In the following, we discuss the results As multiplying the labels of an L(1, 1)- dealing with a subclass of bounded clique- labelling by 2 we obtain an L(2, 1)-labelling, width graphs. the exact algorithm provided in [170] is a 2-approximate algorithm for the L(2, 1)- 4.5.2. Bounded Treewidth Graphs labelling problem. The class of t-trees is recursively defined as Open Problem: To the best of the au- follows: thor’s knowledge, there are no results con- 1. Kt is a t-tree; cerning the L(2, 1)-labelling on graphs of 2. if H is a t-tree, then the graph obtained bounded clique-width greater than two, from H by adding a new node joining to a t-clique (i.e. Kt) of H is a t-tree; so any bound on λ2,1 for these graphs is welcome. 3. all t-trees can be formed with rules 1 and 2. The graphs of clique-width 2 coincide with Any tree is a 1-tree. t-trees are also a the class of cographs, that can be defined subclass of chordal graphs. alternatively as follows. Any subgraph of a t-tree is called partial Let G and H be two graphs with disjoint t-tree. The partial t-trees are a particular node sets. The union of G and H, G H, is case of graphs of bounded clique-width, more ∪ The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 15 precisely, if G is a partial t-tree then cwd(G) labelling problem [129]. 2t+1 + 1 [49]. The minimum value of t for≤ Parameterized complexity of these problems which a graph G is a subgraph of a t-tree is was considered in [66]. It is proved that called the treewidth tw(G) of the graph. See L(0, 1)- and L(1, 1)-labellings are W [1]-hard [19, 154] for surveys on treewidth. Many NP- when parameterized by the treewidth of the hard problems have been shown to be solvable input graph. It shows that the known algorith- in polynomial time on graphs with bounded mic results about graphs of bounded treewidth treewidth. are tight in some sense, i.e. we cannot expect This class is interesting in the wireless algorithms with running time f(tw)nc unless networks context, since pairs of antennas have FPT = W [1]. ******CERCARE DI CAPIRE no interference if their distance is far enough. COS’E’ QUESTO F(TW)NC******* Furthermore, concentrations of antennas are L(h, k)-labelling. In [41], the authors give found in densely populated areas. These an upper bound on the L(h, 1)-numbers of t- areas are connected with one another with a trees, proving that λ (G) (2h 1+∆ t)t. h,1 ≤ − − limited number of edges. Such networks can be represented by a constraint graph with a Open Problem: Upper and lower bounds on tree-like structure [1]. λh,k(G) for graphs of bounded treewidth Fiala and Kratochv´ıl[70] observe that, given are completely unexplored. The relevance of this class of graphs makes this open a fixed span σ, the question ”Is λh,k(G) σ?” can be expressed in MSOL (Monadic Second≤ problem an interesting issue. Order Logic), so this decision problem can be decided in polynomial time for graphs 4.6. Planar Graphs of bounded treewidth. On the contrary, if planar the span is part of the input, the L(2, 1)- A graph G is if and only if it can be labelling problem is NP-complete for graphs drawn on a plane so that there are no edge of treewidth at most two [63]. This result crossings, i.e. edges intersect only at their adds a natural and well studied problem to common extremes. the short list of problems whose computational In many real cases the actual network complexity separates treewidth one from topologies are planar, since they consist of treewidth two. Indeed, usually, the problems communication stations located in a geograph- solvable in polynomial time for trees are also ical area with non-intersecting communication polynomially solvable for graphs of bounded channels [1]. treewidth, though sometimes the extension to The decision version of the L(h, k)-labelling bounded treewidth is not straightforward. problem is NP-complete for planar graphs [151] and even for planar graphs of bounded L(0, 1)-, L(1, 1)- and L(2, 1)-labelling. degree [150]. Bodlaender et al. [20] compute upper bounds L(1, 1)-labelling. The first reference for graphs of treewidth bounded by t proving concerning the L(1, 1)-labelling problem on that λ0,1(G) t∆ t, λ1,1(G) t∆ planar graphs seen as the problem of coloring and λ (G) ≤ t∆+2− t. They give≤ also 2,1 ≤ the square of graphs, is by Wegner [180], who approximation algorithms for the L(0, 1)- gives bounds on the clique number of the , L(1, 1)- and L(2, 1)-labellings running in square of planar graphs. In particular, he gives O(tn∆) time. Nevertheless, two of these an instance for which the clique number is at three problems can be optimally solved: least 3/2∆ + 1 (which is largest possible), in [187] a polynomial time algorithm to and conjectures⌊ ⌋ this to be an upper bound optimally L(1, 1)-label graphs with constant 2 on χ(G ) (i.e. on λ1,1(G) + 1), for ∆ 8. treewidth t is presented, but it applies ≥ He conjectures also that λ1,1(G) ∆+4 dynamic programming and the required time ≤ 8(t+1)+1 for 4 ∆ 7. Moreover, Wegner proves is very high: O(n) O(∆2 ) + O(n3). ≤ ≤ · that λ1,1(G) 7 for every planar graph G of A similar argument would yield the same maximum degree≤ 3, and conjectures that this result for the L(0, 1)-labelling problem. If upper bound could be reduced to 6. the graph with constant treewidth has also constant maximum degree, than O(n) time Open Problem: All three Wegner’s conjec- is sufficient to optimally solve the L(1, 1)- tures remain open, although Thomassen The Computer Journal, Vol. ??, No. ??, ???? 16 T. Calamoneri [172] thought to have established the lat- apply here resulting to an 1.66-approximation ter conjecture. algorithm. Given a planar graph G, Jonas [112] in L(2, 1)-labelling. It is NP-complete his Ph.D. thesis proves that λ (G) 8∆ to decide whether λ2,1(G) r for a 1,1 ≤ 21. This bound is later improved in≤ [173] to− planar bipartite graph of degree r 1 [20]. − λ1,1(G) 3∆ + 4 and in [100] to λ1,1(G) Nevertheless, it seems that the technique used 2∆ + 24.≤ The better asymptotic bound≤ cannot help to show the NP-completeness of λ (G) 9 ∆ + 1 holds for very large values the problem of deciding whether a given planar 1,1 ≤⌊ 5 ⌋ of ∆ (∆ 749) [8]. Borodin et al. [23] graph G has λ2,1(G) r for any odd values ≤ show that≥ we only need ∆ 47 for the bound of r. The authors leave this as an open 9 ∆ to hold. Furthermore,≥ they prove that problem. This problem has been investigated ⌈ 5 ⌉ λ1,1(G) 58 if ∆ 20 and λ1,1(G) ∆+38 in [74] and then definitively closed in [57], if 21 ≤∆ 46.≤ But the better asymptotic≤ where Eggemann, Havet and Noble provide a ≤ ≤ 5 proof of NP-completeness for planar graphs of result is the following: λ1,1(G) 3 ∆ +77 for any planar graph G and λ (G≤⌈) 5⌉∆ + 24 any degree. 1,1 ≤⌈ 3 ⌉ if ∆ 241 [143]. Some of the above results Jonas [112] proves that λ2,1(G) 8∆ ≤ − were≥ obtained by identifying so-called light 13 when G is planar. This bound has structures in planar graphs. The interested been the best one until recently, when the reader can see the survey [104]. general bounds for λh,k found by Molloy and For planar graphs of large girth, better Salavatipour and discussed in the following upper bounds for λ1,1(G) are obtained by have been derived. the results on general h and k in [177], listed below. In [167] the authors show that Open Problem: The Griggs and Yeh’s conjecture is still open for planar graphs λ1,1(G) 5∆ if G is planar. Comparing ≤ with ∆ = 3, while is known to be true for this result with the best known one λ1,1(G) ≤ the other values of ∆. Namely, for ∆ 7 5/3∆ +77, we get that 5∆ is better for any ≥ ∆⌈ 24.⌉ it follows from [100], while Bella et al. [15] ≤ prove it for 4 ∆ 6. Further details on Wang and Lih [177] conjecture that for ≤ ≤ any integer g 5, there exists an integer these results are given below. M(g) such that≥ if G is planar of girth g and L(h, k)-labelling. Van den Heuvel and maximum degree ∆ M(g), then λ ∆. 1,1 McGuinnes [100] show that λ is bounded This conjecture is known≥ to be false for≤g = h,k above by (4k 2)∆+10h+38k 23 for planar 5, 6 and true for g 7 [24, 25]. Nevertheless, graphs for any− positive integers−h and k, such the conjecture is ”almost”≥ true for g = 6, in that h k. This bound implies λ (G) the sense that for ∆ large enough (∆ 8821), ≥ 2,1 ≤ ≥ 2∆+35 and λ1,1(G) 2∆+24. Then, in [143], λ ∆ + 1 if g 6 [56]. ≤ 1,1 ≤ ≤ the result is improved to k 5 ∆ +18h+77k 18 ⌈ 3 ⌉ − Open Problem: It is not known whether an for any positive integers h and k. Observe that analogous statement can hold for planar this latter value is asymptotically better than graphs of girth 5. the previous ones and leads to the best known bounds on λ and λ , of 5 ∆ + 95 and 2,1 1,1 ⌈ 3 ⌉ Open Problem: The problem of tightly 5 ∆ + 77, respectively. ⌈ 3 ⌉ bounding λ1,1 for planar graphs with relatively small ∆ is far to be solved, and Open Problem: Since for a planar graph G it would be a very interesting result. it holds the trivial lower bound λh,k(G) ∆k + h k, it remains an open problem≥ An approximation algorithm for the L(1, 1)- to understand− which is the tight constant labelling problem with a performance guaran- multiplying ∆ in the value of λh,k(G) tee of at most 9 for all planar graphs is given or, at least, of λ2,1(G). This is a very in [152]. For planar graphs of bounded degree, discussed problem, as the large amount of in [129] there is a 2-approximation algorithm. produced literature proves. Furthermore, For planar graphs of large degree (∆ 749) this question gives sense to all the results an 1.8-approximation algorithm is presented≥ in cited above, holding only for very large [8]. The results of [143], given for any h and k, values of ∆. The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 17 In the case h = 2 and k = 1, the bound From the results in [176] it follows that of Van den Heuvel and McGuinnes implies there exist two constants c1 and c2 such that the conjecture of Griggs and Yeh (i.e. that for all planar graphs G without 4- 2 λ2,1 ∆ ) holds for planar graphs with cycles λh,1(G) lies in the interval [∆+h maximum≤ degree ∆ 7. In [15] it is shown 1, ∆+c ], h =1, 2. It is not known which− ≥ h that the conjecture holds for planar graphs are the precise values of c1 and c2. We with maximum degree ∆ = 3. Namely, the remark that c1 and c2 are not bounded authors prove that every6 planar graph with when planar graphs G are allowed to have maximum degree 4, 5, 6, respectively, has 4-cycles. an L(2, 1)-labelling with span at most 16, 25, 32, respectively, and the proof in the case of 4.6.1. Outerplanar and l-Outerplanar maximum degree 4 is computer-assisted. Graphs The L(h, k)-labelling problem is also studied A graph is outerplanar if it can be embedded on planar graphs with conditions on their in the plane so that every node lies on the girth. More precisely, in [177] the following boundary of the outer face. bounds are proven: A graph G is l-outerplanar if for l = 1, if g(G) 7, then λh,k(G) (2k 1)∆ + G is outerplanar and for l > 1 G has a − 4h +4k ≥ 4; ≤ − planar embedding such that if all nodes on − if g(G) 6, then λh,k(G) (2k 1)∆ + the exterior face are deleted, the connected − 6h + 12k≥ 9; ≤ − components of the remaining graph are all − if g(G) 5, then λh,k(G) (2k 1)∆ + (l 1)-outerplanar. − 6h + 24k≥ 15. ≤ − −It can be determined in polynomial time − Observe that, since ∆+8 ∆2 when ∆ 4, whether G is outerplanar and whether it is l- ∆+15 ∆2 when ∆ 5,≤ and ∆+21 ≥∆2 outerplanar. ≤ ≥ ≤ when ∆ 6 the conjecture by Griggs and Yeh L(1, 1)-labelling. Any l-outerplanar graph holds for≥ planar graphs with g(G) 7 and ≥ has a treewidth of at most 3l 1 [18]. ∆ 4 or g(G)=6 and ∆ 5, or g(G)=5 Moreover, outerplanar graphs are− series- and≥ ∆ 6. ≥ ≥ parallel graphs, and series-parallel graphs are Furthermore, if the degree is sufficiently exactly partial 2-trees. Thus, applying the large (very large, indeed), a better bound result of [187] for bounded treewidth graphs, can be provided: every planar graph of girth we get that any l-outerplanar and outerplanar g 7 has an L(h, k)-labelling of span at ≥ h graph can be optimally L(1, 1)-labelled in most 2h + ∆k 2 if ∆ 190 + 2 [55]. 3 − ≥ ⌈ k ⌉ O(n ) time. Since the optimal span of an L(h, 1)-labelling In [35] a linear time algorithm for optimally of an infinite ∆-regular tree is 2h + ∆ 2, the − L(1, 1)-labelling any outerplanar graphs of obtained bound is the best possible for any degree ∆ 7 with at most ∆ + 1 colors h 1 and k = 1. In [176], the authors study ≥ ≥ is presented. When ∆ 6 the required L(h, k)-labellings for planar graphs without 4- number of colors is anyway≥ 11. This cycles, as such graphs possess some interesting algorithm first executes a special traversal of properties. The authors prove the following an embedding of the graph (ordered breadth bound for planar graphs without 4-cycles: first search), then it labels the nodes in a λh,k min (8k 4)∆ + 8h 6k 1, (2k greedy fashion starting from the root of the 1)∆ +≤ 10h +{ 84k− 47 for any− h, k− 1 that− − } ≥ resulting spanning tree and following a level is asymptotycally better than any previous by level order. The optimality proof exploits bound on this subclass of planar graphs. As an some strong properties of the non-tree edges. immediate consequence, it follows that λ 2,1 ≤ Later, Agnarsson and Halld´orsson [7] derive min 4∆+9, ∆+57 and λ1,1 min 4∆ + optimal upper bounds on λ1,1 for outerplanar 1, ∆{ + 47 . Hence,} the Griggs≤ and{ Yeh } graphs of small degree (∆ < 7) proving that conjecture on λ2,1 and the Wegner conjecture λ1,1 ∆+2if∆= 2and λ1,1 ∆ + 1 if on λ1,1 hold for planar graphs without 4-cycles ≤ ≤ ∆=3, 4, 5 and λ1,1 ∆ if ∆ = 6. having ∆ 9 and ∆ 96, respectively. ≤ ≥ ≥ L(2, 1)-labelling. As outerplanar graphs Open Problem: For any graph G, it is well are graphs of treewidth 2, from the result in known that λ ∆ and λ ∆ + 1. [20] for bounded treewidth graphs, we have 1,1 ≥ 2,1 ≥ The Computer Journal, Vol. ??, No. ??, ???? 18 T. Calamoneri an immediate upper bound, i.e. λ2,1(G) Wang and Luo [178] prove some bounds on 2∆ + 4. Jonas [112] proves the slightly better≤ small degree outerplanar graphs: λ (G) 2,1 ≤ bound λ2,1(G) 2∆ + 2. Providing a coloring ∆+4if∆=3and λ2,1(G) ∆ + 5 if ∆ = 4, algorithm, in [20],≤ the authors improve this so improving the previous bounds≤ in these two bound to λ2,1(G) ∆+8 for any outerplanar cases. Furthermore, these author provide a graph G, but they≤ conjecture that the tightest graph with ∆ = 4 disproving the previous bound could be ∆ + 2. Calamoneri and conjecture as it needs 8 colors (see Figure 8). Petreschi [35] prove this conjecture when the Some of the previous bounds for small values input outerplanar graph has maximum degree of ∆ are improved in [28]. In particular, ∆ 8, and they provide a linear time the authors prove that λ2,1(G) 6 = algorithm≥ that guarantees an L(2, 1)-labelling ∆ + 3 for each degree 3 outerplanar≤ graph. of G with a number of colors far at most one Furthermore, by experimental techniques, from optimum. The algorithm is analogous to they improve the lower bounds for 4 ∆ 6 that one presented for the L(1, 1)-labelling. showing some graphs requiring ∆ + 3≤ colors.≤ Open Problem: The authors of [35] conjec- Open Problem: It remains an open problem ture that this algorithm is optimal; if this to close the gap between upper and lower is true, the L(2, 1)-labelling problem on bounds for 4 ∆ 7. ≤ ≤ outerplanar graphs would be polynomi- The upper bounds on λ (G) can be ally solvable. The question is still open, 2,1 slightly improved if the outerplanar graph G and particularly interesting because out- is triangulated [20, 35]. erplanar graphs are perfect and such a re- sult would help to understand the rela- Open Problem: No results are known about tionship between the hardness of the ver- the general L(h, k)-labelling problem on tex coloring and of the L(2, 1)-labelling outerplanar graphs. problem. 4.6.2. -Minor Free Graphs For outerplanar graphs with smaller values K4 A graph G has a graph H as a minor if H of ∆, in [35] it is guaranteed λ2,1(G) 10, improving anyway the bound of ∆+8, but≤ the can be obtained from a subgraph of G by -minor authors conjecture that the bound ∆+2 holds contracting edges, and G is called H free for any outerplanar graph of degree ∆ 4. if G does not have H as a minor. A series-parallel Differently, in the special case ∆ = 3,≥ it is graph is called graph if G can shown that there exists an infinite class of be obtained from K2 by applying a sequence outerplanar graphs needing ∆ + 4 colors and of operations, where each operation is either an algorithm using at most ∆ + 6 colors for to duplicate an edge (i.e. replace an edge with any degree 3 outerplanar graph is presented. two parallel edges) or to subdivide an edge (i.e. replace an edge with a path of length 2). A graph G is K4-minor free if and only if each block of G is a series-parallel graph. It is well known [44] that a graph G is outerplanar if 6 and only if G is K4-minor free and K2,3-minor free. Thus, the class of K -minor free graphs 3 0 4 is a class of planar graphs that contains both 4 6 outerplanar and series-parallel graphs. 2 1 2 3 L(1, 1)-labelling. If G is a K4-minor free 5 graph, then λ1,1(G) ∆ + 2 if 2 ∆ 3 and λ (G) 3 ∆ otherwise≤ [135].≤ The≤ result 0 6 4 0 1,1 2 is best possible,≤ ⌊ ⌋ as witnessed by the following 3 7 graphs: for ∆ = 2, cycle C5, having λ1,1 = 4; − for ∆ = 3 let G be the graph consisting of FIGURE 8. A counterexample outerplanar − graph with ∆ = 4 and λ2,1 = ∆ + 3. three internally disjoint paths joining two nodes x and y, where two of the paths are The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 19 of length 2 and the third one is of length 3; 4.7. Graphs with an Intersection this graph has λ1,1 = 5 (see Fig. 9.a); Model for ∆ = 2d 4 let G be the graph 2d For a given set M of objects (for which − consisting of ≥d internally disjoint paths intersection makes sense), the corresponding joining x and y, d internally disjoint paths intersection graph G is the undirected graph joining x and z, and d internally disjoint whose nodes are objects and an edge connects paths joining y and z; all these paths are two nodes if the corresponding objects of length 2, except one path joining x and intersect. M is called the model of G with y of length 1, and one path joining x and z respect to intersection. of length 1. The degree of such a graph is Depending on the nature of the object, 2d and its λ is 3d (see Fig. 9.b where d 1,1 many interesting classes can be defined. is 3); for ∆=2d +1 5, let G be obtained 2d+1 4.7.1. Disk Graphs and -Civilized − from G by adding≥ a new path of length (r, s) 2d Graphs 2 joining y and z; this graph has λ = 1,1 A disk graph is the intersection graph of a 3d + 1. set of disks in the plane, where each disk y is uniquely determined by its center and its diameter. The class of disk graphs is very wide and interesting, and it includes classes as, for x y instance, planar graphs. When all disks are of the same diameter the graph is called unit disk x graph. The disk graph and unit disk graph z recognition problem is NP-hard [27, 102]. Hence, labelling algorithms that require the FIGURE 9. K4-minor free graphs having λ1,1 = corresponding disk graph representation are 3 ⌊ 2 ∆⌋ = n. substantially weaker than those which work only with graphs. For each fixed pair of real values r > 0 and L(2, 1)-labelling. It is NP-complete to s > 0, a graph G belongs to the class of the decide whether λ (G) σ, where σ is h,k -civilized graphs part of the input, if G is≤ series-parallel [63], (r, s) if there exists a positive integer d 2 such that the intersection as a consequence of the results on bounded model is a≥ set of spheres of Rd, the centers treewidth graphs, as series-parallel graphs are exactly partial 2-trees. Of course, the of intersecting spheres are at distance r and the distance between any two centers≤ is same result holds for K4-minor free graphs, a superclass of series-parallel graphs. s [171]. In the following, planar (r, s)- civilized≥ graphs (i.e. with d = 2) will L(h, k)-labelling. Wand and Wang [179] be treated; however, all the results can be show that every K4-minor free graph G with extended directly to civilized graphs of higher maximum degree ∆ has an L(h, k)-labelling, dimension. h + k 3, with span at most 2(2h 1) + Note that the class of (r, s)-civilized graphs ≥ 3 − (2k 1) 2 ∆ , generalizing the previous result includes disk graphs whenever there is a on the− L⌊(1, 1)-labelling.⌋ It is natural to wonder (fixed) minimum separation between the whether the bound on λh,k is optimal, as the centers of any pairs of circles. one on λ1,1. This is not the case, as proved in The previously described classes of graphs [126], where it is shown that for every h 1 are considered as reasonable models for several ≥ there exist ∆0 such that every K4-minor free classes of packet radio networks. To see this, graph with maximum degree ∆ ∆0 has consider packet radio networks in which the ≥ 3 an L(h, 1)-labelling with span at most 2 ∆ range of any transmitter can be considered and this bound cannot be further decreased.⌊ ⌋ as a circular region with the transmitter at This result translates to L(h, k)-labelling, with the center of the circle; let r be the radius k > 1 providing an upper bound for λh,k of of the region corresponding to a transmitter’s k 3 ∆ . maximum range. Further, it is natural to ⌊ 2 ⌋ The Computer Journal, Vol. ??, No. ??, ???? 20 T. Calamoneri assume a minimum separation s between any a linear function of the size of the maximum pair of transmitters, otherwise the equipment clique in G. carrying the transmitters cannot be placed. In [168] the first known upper bound for unit Clearly, the graphs that model such networks disk graphs in terms of ∆ is shown: λ2,1(G) 4 2 ≤ belong to the class of (r, s)-civilized graphs. 5 ∆ + 2∆. In many other realistic situations, the ratio of maximum to the minimum transmitter range Open Problem: This latter upper bound is is not fixed; in such cases disk graphs are more far from being tight, indeed consider as realistic [129]. an example the triangular grid G6: it is a unit disk graph of maximum degree 6, L(1, 1)-labelling. The decision version of its span is 8, but the value of this upper the problem is NP-complete for unit disk bound is 40. graphs. Krumke, Marathe and Ravi [129] give a 2-approximation algorithm for the L(1, 1)- L(h, k)-labelling. In [62] it is also labelling problem for (r, s)-civilized graphs. studied also the more general L(h, k)-labelling The performance guarantee of the algorithm problem on disk graphs (in fact the even more is independent of the values of r and s. An general L(p ,p ,...p )-labelling problem) and approximation algorithm with a performance 1 2 r it is presented an approximation algorithm guarantee of 14 for disk graphs is given in whose performance depends on the diameter [162]. It has been later shown [137] that ratio σ, i.e. the ratio between the biggest and the performance guarantee of 14 can also be the smallest diameters of the set of disks. achieved using the greedy paradigm. The performance ratio has been improved to 13 (to 12 if the radii are quasi-uniform) [174], by 4.7.2. Chordal Graphs means of FIRST-FIT algorithms. In [162, 174] A graph is chordal (or triangulated) if and two approximation algorithms for unit disk only if it is the intersection graph of subtrees graphs with a performance guarantee of 7 are of a tree. An equivalent definition is the presented. Finally, in [174] the authors prove following: a graph is chordal if every cycle of that for unit disk graphs whose nodes lie in length greater than three has a chord. Chordal √3 graphs have been extensively studied as a a horizontal strip of heigth 2 the L(1, 1)- labelling problem can be solved in polynomial subclass of perfect graphs [105]. time. The reason is that such graphs are co- An n-sun is a chordal graph with a comparability graphs, that are perfect graphs Hamiltonian cycle x1, y1, x2, y2, ... xn, with the property that their square graphs yn, x1 in which each xi is of degree exactly are still co-comparability graphs, and hence 2. A sun-free-chordal (respectively odd-sun- optimally L(1, 1)-labellable in polynomial time free-chordal) graph is a chordal graph which (see Subsection 4.9.8). contains no n-sun with n 3 (respectively odd n 3) as an induced subgraph.≥ Sun-free- ≥ Open Problem: Is it possible to improve the chordal graphs are also called strongly chordal performance ratios of 13 and 7 for disk graphs and are particularly interesting as they and unit disk graphs, respectively? include directed path graphs, interval graphs, unit interval graphs and trees. Also strongly chordal graphs can be defined as intersection -labelling. L(2, 1) Fiala, Fishkin and Fomin graphs of subtrees with certain properties of a [62] explore the L(2, 1)-labelling problem on tree. A graph G is called a block graph if each disk and unit disk graphs. For the first block of G is a complete graph. The class of class of graphs they provide an approximation block graphs includes trees and is a subclass algorithm having performance ratio bounded of strongly chordal graphs. by 12. For the second class, they present a robust labelling algorithm, i.e. an algorithm L(1, 1)-labelling. In [6] it is proven that does not require the disk representation that the L(1, 1)-labelling problem on chordal and either outputs a feasible labelling, or graphs is hard to approximate within a factor 1 ǫ answers the input is not a unit disk graph. Its of n 2 − , for any ǫ > 0, unless NP-problems performance ratio is constant and bounded by have randomized polynomial time algorithms. 32/3. In both cases, λ2,1(G) is bounded by The authors match this result with a simple The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 21 O(√n)-approximation algorithm for L(1, 1)- free chordal graph, then λ (G) h∆ and if h,1 ≤ labelling chordal graphs. G is strongly chordal then λh,1(G) ∆+(2h 2)(χ(G) 1) [41]. Also for this problem≤ it− is L(2, 1)-labelling. To decide whether − 3/2 proven that λh,k(G)= O(∆ (2k 1)) [124]. λ2,1(G) n is NP-complete if G is a chordal − graph [20];≤ this can be proven by means 4.7.3. Interval Graphs of a reduction from HAMILTONIAN PATH interval graph and exploiting the notion of complement of a An is an intersection graph graph. whose model is a set of intervals of the real The L(2, 1)-labelling for chordal graphs has line. unit interval graphs been first investigated by Sakai [158] in order The class of is a subclass of interval graphs for which all the intervals to approach the general conjecture λ2,1(G) ∆2. The author proves that chordal graphs≤ are of the same length, or equivalently, for satisfy the conjecture and more precisely that which no interval is properly contained within 1 2 another. λ2,1(G) 4 (∆ + 3) . Chang≤ and Kuo [42] study upper bounds Interval graphs are used to model wireless networks serving narrow surfaces, like high- on λ2,1(G) for odd-sun-free-chordal graphs and strongly chordal graphs and prove that ways or valleys confined by natural barriers λ (G) 2∆ if G is odd-sun-free-chordal and (e.g. mountains or lakes). 2,1 ≤ λ2,1(G) ∆+2χ(G) 2 if G is strongly L(1, 1)-labelling. Interval and unit- ≤ − chordal. Although a strongly chordal graph interval graphs are perfect; furthermore, is odd-sun-free-chordal, the upper bounds are interval graphs are closed under powers and incomparable. The result on strongly chordal the square of a unit-interval graph is still a graphs is a generalization of the result that unit-interval graph [153]. It follows that the λ (T ) ∆+2 for any non trivial tree T . The L(1, 1)-labelling problem on interval and unit- 2,1 ≤ authors conjecture that λ2,1(G) ∆+ χ(G) interval graphs is polynomially solvable. A ≤ for any strongly chordal graph G. All the linear time algorithm for finding an optimal previous results for chordal graphs have been L(1, 1)-labelling of interval graphs is presented improved by the result in [124], stating that in [17]. the λ - and λ -numbers are both O(∆3/2) 1,1 2,1 L(2, 1)-labelling. Sakai [158] proves that for this class of graphs, and that there exists a 2χ(G) 2 λ (G) 2χ(G) for unit interval chordal graph G such that λ (G) = Ω(∆3/2). 2,1 2,1 graphs.− In≤ terms of ∆,≤ as χ(G) ∆+1, the Some subclasses of chordal graphs have upper bound becomes λ (G) ≤ 2(∆ + 1), been investigated and, restricted to them, the 2,1 and this value is very close to≤ be tight, as general bounds on λ of chordal graphs have 2,1 the clique K , that is an interval graph, has been improved. In particular, in [148] it has n λ (K )=2(n 1) = 2∆. been proved that for chordal bipartite graphs 2,1 n − λ2,1 4∆ 1, hence the Griggs and Yeh’s L(h, k)-labelling. In [17] the authors conjecture≤ is− true for chordal bipartite graphs present a 3-approximate algorithm for L(h, 1)- with ∆ = 3. Furthermore, a graph is a block labelling interval graphs. In the special case of graph if6 it is connected and every maximal 2- unit interval graphs, the same approximation connected component is a clique. As block ratio holds for the L(h, k)-labelling problem. graphs are strongly chordal, all the results An L(h, k)-labelling algorithm for interval for this latter class holds for the former one, graphs with span at most max(h, 2k)∆ is i.e. λ2,1 2∆ and λ2,1 ∆+2χ 2 = provided in [33]; this span can be slightly ∆+2ω 2,≤ as χ = ω for a block≤ graph. In− [147] improved under some constraints that the − these results are improved to λ2,1 ∆+ ω. graph has to respect. In the same paper, it Nevertheless, if ∆ 4, ω = 3 and G≤does not is proved that the classical greedy algorithm contain a certain subgraph≤ with 7 nodes and guarantees a span never larger than min((2h+ 9 edges, then λ2,1 6 [22]. 2k 2)(ω 1), ∆(2k 1)+(ω 1)(2h 2k)), ≤ where− ω is− the dimension− of the− larger− clique L(h, k)-labelling. As a generalization of in the graph. the result known for h = 2, if G is a chordal graph with maximum degree ∆, then Open Problem: It is still not known λ (G) 1 (2h + ∆ 1)2; if G is an odd-sun- whether the decisional version of the h,1 ≤ 4 − The Computer Journal, Vol. ??, No. ??, ???? 22 T. Calamoneri L(h, k)-labelling problem is NP-complete of a bipartite graph is the number of nodes on interval graphs or not. Concerning this in a maximum biclique.) Since λh,k(G) problem, the feeling of the author is that h + k(bc(G) 2) for any bipartite graph, this≥ it is NP-complete, even for unit-interval algorithm guarantees− a number of colors that graphs. is at most h 1 far from optimal. Open Problem: From the results for − interval graphs, the authors of [33] deduce 4.7.5. Split Graphs a result on circular arc graphs, i.e. A graph G is a split graph if and only if G is the intersection graphs whose model is a set intersection graph of a set of distinct substars of intervals in a circle. The approach they of a star. Alternatively, a split graph is a graph follow is: i. to consider a clique in the G of which node set can be split into two sets graph whose remotion gives an interval K and S, such that K induces a clique and S graph, ii. to label the interval graph, iii. an independent set in G. All split graphs are to insert again the clique labeling it with chordal. further labels. In this way, it is possible to guarantee λh,k min((3h +2k 2)ω L(0, 1)-and L(1, 1)-labelling. Bodlaender (2h +2k 2), ∆(2≤k 1) + ω(3h − 2k) − et al. [20] prove that it is NP-complete − − − − (2h 2k). The interest of this result is to decide both whether λ0,1(G) 3 and − ≤ that it is the first one dealing with circular whether λ1,1(G) r when r is given in ≤ arc graphs, but it should be possible to input and G is a split graph. This also improve it. An interesting open problem implies NP-completeness of the problems to is to provide tight upper and lower bound decide the λ0,1- and λ1,1-numbers for chordal on λh,k for circular arc graphs. graphs. In [6] it is proven that the L(1, 1)- labelling problem on split graphs is hard to 1/2 ǫ 4.7.4. Permutation Graphs approximate within a factor of n − , for any An intersection model of straight lines between ǫ > 0, unless NP-problems have randomized two parallel lines describes permutation graphs polynomial time algorithms. as follows: let , be two parallel lines in L1 L2 L(2, 1)-labelling. As split graphs are the plane and label n points by 1,2, ...,n chordal, the results stated for chordal graphs (not necessarily in this order) on 1 as well hold for interval graphs. Moreover, split as on . The straight lines L Lconnect i L2 i graphs are the first known class of graph for on 1 with i on 2. = L1,...,Ln is L L L { } which λ2,1 is neither linear nor quadratic in the intersection model of the corresponding ∆. Namely, in [20] it is presented an algorithm permutation graph. L(2, 1)-labelling G with at most ∆1.5 +2∆+3 The name permutation graph comes from colors, and it is shown that there exist split the fact that the points on , can be L1 L2 graphs for which this bound is tight. Similar seen as a permutation π = π1,...πn and bounds are obtained also for λ0,1 and λ1,1. (i, j) E(G) if and only if i{and j form} an ∈ inversion in π. Open Problem: Split graphs and chordal L(0, 1) ,L(1, 1)- and L(2, 1)-labelling. graphs (see Subsection 4.7.2) represent In [20]− it is described an approximation the only known class for which λ2,1 is algorithm for L(h, 1)-labelling (h = 0, 1, 2) neither linear nor quadratic in ∆. It a permutation graph in O(n∆) time; it remains an open problem to understand if there exist other classes of graphs guarantees the following bounds: λ0,1(G) ≤ whose λ2,1-number has this property. A 2∆ 2, λ1,1(G) 3∆ 2 and λ2,1(G) 5∆ 2. − ≤ − ≤ − characterization of the class of graphs 1.5 L(h, k)-labelling. For those permutation having λ2,1 = Θ(∆ ) would be a graphs that are also bipartite, there exists probably hard but very interesting target. a polynomial L(h, k)-labelling approximation algorithm [11] that guarantees to use at most 4.8. Hypercubes and Related Net- 2h 1+k(bc(G) 2) colors, where bc(G) is the works biclique− number− of G. (In a bipartite graph, a subset of nodes is a biclique if it induces a The n-dimensional hypercube Qn is an n- complete bipartite graph. The biclique number regular graph with 2n nodes, each having a The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 23 binary label of n bits (from 0 to 2n 1). Two λ (Q ) n + 4 for n = 8 and n = 16, − 2,1 n ≥ nodes in Qn are adjacent if and only if their respectively [111]. Furthermore, λ2,1(Qn) binary labels differ in exactly one position. 2n +1 for n 5 [89]. The same authors also≤ The n-dimensional hypercube Q can also determine λ ≥(Q ) for n 5 and conjecture n 2,1 n ≤ be defined as the cartesian product of n K2 that the lower bound n + 3 is the actual value graphs. The more general cartesian product of λ2,1(Qn) for n 3. Using a coding theory 2 2 2 ≥ Kn1 Kn2 ... Knd of complete graphs is method, the upper bound is improved by 1 in called a Hamming graph, where ni 2 for each [182], where it is proven that it ranges from i =1,...,d. ≥ n + 1 + log n to 2n, depending on the value ⌊ 2 ⌋ The N-input Butterfly network BN (with N of n. Furthermore, lim inf λ2,1(Qn)/n = 1. power of 2) has N(log2 N + 1) nodes. The In [75] an L(2, 1)-labelling algorithm of Qn nodes correspond to pairs (i, j), where i (0 is described: exploiting a coding theoretic i log n what concerns CCCn, the authors provide a L(0, 1)-labelling. λ0,1(Qn) 2⌈ ⌉ and ≤ labelling ensuring λ2,1(CCCn) 6, that is there exists a labelling scheme using such a 1 far from optimal, and they experimentally≤ number of colors. This labelling is optimal k verify that there exist some values of n (e.g. when n = 2 for some k, and it is a 2- n = 5) requiring a 6 colors labelling. approximation otherwise [173]. L(1, 1)-labelling. In [173] an L(1, 1)- Open Problem: The approach presented in [32], consisting in shrinking some labelling scheme of the hypercube Qn is described in the context of optical cluster cycles of the networks and in reducing based networks. A different approach is used to label these simpler graphs instead in [52] with respect to the problem of data of the complete networks, seems to be distribution in parallel memory systems: it promising: it is very general both because log n +1 it can be applied to many multistage is presented an algorithm that uses 2⌊ ⌋ colors, and requires O(n) time and space, networks, and because it works for every improving the previously known results. In value of h and k. Nevertheless, it needs to be refined: first of all, the both papers, the upper bound on λ1,1(Qn) is a 2-approximation, that is conjectured to be authors themselves realize that sometimes the best possible. the reduced graph needs more colors than the whole network, because the L(2, 1)-labelling. For the n-dimensional reduced graph typically has degree higher hypercube Q , λ (Q ) n + 3 and than the original network. Secondly, n 2,1 n ≥ The Computer Journal, Vol. ??, No. ??, ???? 24 T. Calamoneri 1st Row - 000 1st Row - 000 2nd Row - 001 2nd Row - 001 3rd Row - 010 3rd Row - 010 4th Row - 011 4th Row - 011 5th Row - 100 5th Row - 100 6th Row - 101 6th Row - 101 7th Row - 110 7th Row - 110 8th Row - 111 8th Row - 111 a. b. FIGURE 10. a. The classical representation of an 8-input Butterfly network; b. The bidimensional layout of a 3-dimensional CCC network.) the main bottleneck of this method is 4.9.1. Diameter 2 Graphs that the reduced graph must be labelled A diameter 2 graph is a graph where all nodes using exhaustive methods; nevertheless, have either distance 1 or 2 each other. it should be relatively easy to design algorithms to efficiently label the reduced L(1, 1)- and L(2, 1)-labelling. Intuitively, graphs, that are constituted by two diameter 2 graphs seem to be a particularly parallel cycles joined by a very regular set feasible class to efficiently solve the L(h, k)- of edges. labelling problem. On the contrary, while it is easy to see that the λ1,1-number for these graphs is n 1, Griggs and Yeh [89] prove − L(h, k)-labelling. Zhou [186] proves that that the L(2, 1)-labelling problem is NP-hard. s s t They prove also that λ ∆2 and state λh,k(Qn) 2 max k, h/2 +2 − min h 2,1 ≤ · { ⌈ ⌉} · { − that this upper bound is sharp≤ only when k, h/2 h, where h k 1, s = 1+ log2 n and⌊ t =min⌋}− 2s n 1,s≥ . In≥ particular,⌊ if 2k ⌋ ∆=2, 3, 7 and, possibly 57 because a diameter { − − s } s t ≥ 2 graph with n = ∆2 + 1 can exist only if h, then λh,k(Qn) 2 k+2 − (h k) h leading ≤s s t − − ∆ is one of these numbers (see more details to λ2,1(Qn) 2 + 2 − 2. The proof of this theorem≤ gives rise to a− systematic way of on these graphs below). Since the diameter is 2, all labels in V must be distinct. Hence, generating L(h, k)-labellings of Qn which use s λ(G) n 1 = ∆2 and therefore the equality 2 labels and have span equal to the right-side ≥ − of the previous formula. For Hamming graphs, holds. the same authors show an upper bound on λh,k L(h, k)-labelling. In [121] bounds for the for special values of ni, i = 1,...,d, and this λh,1-number are presented, for all h 2. In bound is optimal when h 2k. ≥ 2 ≤ particular, it is proven that λh,1(G) ∆ + (h 2)∆ 1 if G is a diameter 2 graph≤ with − − 4.9. Other Graphs maximum degree ∆ 3 and n ∆2 1. Since a diameter 2 graph≥ can have≤ at most− 2 In this section there are collected some classes ∆ + 1 nodes, and, with the exception of C4, of graphs for which very few results appear there are no diameter 2 graphs with maximum in the literature, and they are not enough to degree ∆ and ∆2 nodes [59], it just remains to justify a devoted section. It is amazing that investigate the graphs with n = ∆2 +1. There the L(h, k)-labelling problem appears more are merely four such graphs; they are regular tricky just for some very studied classes of and have ∆ = 2, 3, 7 and 57, respectively: graphs, included in this collection, that are ∆ = 57: it is neither known whether very relevant from the theoretical point of view − such a graph exists; this hypothetical graph and that have many interesting properties. is called Aschbacher graph. Junker [114] The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 25 0 L(h, k)-labelling. The λh,1-number of an r-regular graph is at least 2h + r 2 [78]. 2+h In [81], Georges and Mauro prove− that the 3+h 1+h λh,k-number of any r-regular graph G is no 3+2h 1 less than the λh,k-number of the infinite r- regular tree (see Section 4.4). Then, they define a graph G to be (h,k,r)-optimal if and only if the equality holds, they consider 2 2+2h the structure of (h,k,r)-optimal graphs for h/k > r and show that (h,k,r)-optimal graph are bipartite with a certain edge-coloring 3 1+2h property. Finally, the same authors determine FIGURE 11. An L(h, 1)-labelling of Petersen the exact λ1,1- and λ2,1-numbers of prisms. graph. More precisely, for n 3, the n-prism Pr(n) is the graph consisting≥ of two disjoint n- cycles v0, v1,...,vn 1 and w0, w1,...,wn 1 − − strongly conjectures that there is no such and edges vi, wi for 0 i n 1. { } ≤ ≤ − 2 a graph, and we will not consider it for Observe that Pr(n) is isomorphic to Cn P2. obvious reasons; In [80] it is proven that λ2,1(Pr(n)) is equal to 5 if n 0mod 3 and to 6 otherwise, improving ∆ = 2: the cycle of length 5: λh,1(C5)=2h ≡ − [78]; the result in [108], and that ∆ = 3: the Petersen graph P : λ2,1(P )=9 − 3 if n 0 mod 4 [89]; for h 3 λh,1(P )=3+2h (see Figure ≡ 11) [121]; ≥ λ1,1(Pr(n)) = 5 if n =3, 6 ∆ = 7: the Hoffman-Singleton graph HS: 4 otherwise. − max 49, 3h λ (HS) 19 + 3h h,1 Open Problem: Regular graphs seem to [121]{ for h }10; ≤ when h =≤ 10 it holds be particularly relevant for the L(h, k)- λ (HS) =≥ 49 that of course is optimal h,1 labelling questions (notice that most of (as n = 50 and HS is a diameter 2 graph). the graphs shown as extremal cases are Open Problem: Since for h h′ regular graphs), so they are worth being ≥ every L(h, 1)-labelling is also a proper studied more deeply. L(h′, 1)-labelling, it holds λh,1(HS)= 49 for ∆ 10. An interesting ≤ 4.9.3. Bipartite Graphs open problem is to improve the upper Bipartite graphs are graphs with χ(G) 2. bound for h > 10. Another question ≤ Nevertheless, their λh,k-number can be very is whether 10 is the highest value for large, as shown in the following. h such that λh,1 = 49. Before detailing the known results for bipartite graphs, we recall some definitions. A 4.9.2. Regular Graphs bipartite graph G = (U V, E), with U = n ∪ | | 1 An r-regular graph is a graph in which all and V = n2, is called a chain graph if there | | nodes have degree exactly r. Although they exists an ordering u1,...,un1 of U and an constitute an interesting class of graphs, they ordering v1,...,vn of V such that N(u1) 2 ⊆ have not been very studied from the L(h, k)- ... N(un ) and N(v1) ... N(vn ), ⊆ 1 ⊆ ⊆ 2 labelling point of view. Indeed, to the best of where N(x) is the set of all adjacent nodes the author’s knowledge, only three papers deal of x. A subset of nodes of a bipartite graph with regular graphs. is a biclique if it induces a complete bipartite subgraph; the maximum order of a biclique of L(2, 1)-labelling. For every r 3, it is NP- G is denoted by bc(G). complete to decide whether an r-regular≥ graph admits an L(2, 1)-labelling of span (at most) L(0, 1)-labelling. Bipartite graphs may 2 λ2,1 = r + 2 [69]. The result is best possible, require λ0,1 = Ω(∆ ), indeed there exist 2 since no r-regular graph (for r 2) allows an bipartite graphs with λ ∆ [20]. Of ≥ 0,1 ≥ 4 L(2, 1)-labelling of span r + 1. course the same bound holds for each λh,1, The Computer Journal, Vol. ??, No. ??, ???? 26 T. Calamoneri h 1. Later this lower bound has been the value of the minimum span is [34]: improved≥ by a constant factor of 1 in [9]. 4 k (∆ 1)k if h 2 − k ≤ L(1, 1)-labelling. The decision version of λh,k(G)= (∆ 2)k +2h if 2 h k (∆ − 1)k + h if h ≤ k ≤ the L(1, 1)-labelling problem is NP-complete − ≥ even for 3-regular bipartite graphs using 4 colors [47]. In [6] it is proven that the and, of course, these latter values match with L(1, 1)-labelling problem on bipartite graphs the previous ones in the special case n1 = ∆, is hard to approximate within a factor of n2 = 1 and k = 1. 1/2 ǫ n − , for any ǫ > 0, unless NP-problems have randomized polynomial time algorithms. 4.9.4. Cayley Graphs Cayley graphs of the group Γ relative to the L(2, 1)-labelling. Since the general upper finite group generating set S is the labeled 2 directed graph G = (V, E) for which V = Γ bound λ2,1(G) ∆ + ∆ 2 [84] holds also ≤ − and E = (u,u ): u V,s S , where the for bipartite graphs of any degree ∆ and the { s ∈ ∈ } edge (u,us) is labeled s. In other words, there lower bound on λ0,1 holds a fortiori for λ2,1, it 2 is an edge labeled s between two nodes of G if follows that λ2,1(G) = Θ(∆ ) for this class of graphs. one is obtained from the other through right multiplication by s. Note that if S = n, then In [20], the authors prove that the decisional | | version of the L(2, 1)-labelling problem is NP- the undirected graph underlying the Cayley graph G is 2n-regular if for all s,s′ S, complete for planar bipartite graphs. ∈ ss′ = 1, hence the upper bounds on regular For the subclass of chain graphs, the 6 L(2, 1)-labelling problem can be optimally graphs hold for Cayley graphs, too. Some authors study the L(h, k)-labelling solved in linear time and λ2,1(G) = bc(G) problem on this class of graphs, when varying [148]. Furthermore, λ2,1(G) 4∆ 1 for a chordal bipartite graph G [148].≤ In− the same the group Γ. The reader may refer to [186] for paper, the L(2, 1)-labelling problem of several abelian groups, and to [12] for more general subclasses of bipartite graphs is studied, such groups. These results contain as a special case as bipartite distance hereditary graphs and the L(2, 1)-labelling of the square grid. perfect elimination bipartite graphs. 4.9.5. Unigraphs L(h, k)-labelling. In [92] the L(h, k)- Unigraphs are graphs uniquely determined by labelling problem is considered even on their own degree sequence up to isomorphism bipartite graphs, and it is proven that the and are a superclass including matrogenic, simplest approximation algorithm, i.e. the matroidal, split matrogenic and threshold one based on First Fit strategy, guarantees graphs. In this section we will deal with all a performance ratio of O(min(2∆, √n)), and these classes of graphs. The interested reader this is tight within a constant factor in view of can find further information related to these 1/2 ǫ classes of graphs in [138]. the n − -hardness result. On the contrary, exact results can be achieved if the bipartite An antimatching of dimension h of X onto Y is a set A of edges such that M(A) = X graphs are complete. Indeed, in the special × case of k = 1, given a complete bipartite graph Y A is a perfect matching of dimension h of X −onto Y . Agraph G = ( v , v ,...v , ) is a G = (U V, E), where U = n1 and V = n2, 1 2 k ∪ | | | | null graph if its edge set is{ empty, irrespective} ∅ n1 n2 [88]: ≥ of the dimension of the node set. A split graph G with clique K and stable max(n 1,n 1+ h) 1 − 2 − set S is matrogenic (Fig. 12.a) if and only if if 0 h 1 ≤ ≤ 2 the edges of G can be colored red and black so (2n2 1)h + max(n1 n2 1+ h, 0) λh,1(G)= − − − that [140]: if 1 h 1 2 ≤ ≤ a.The red subgraph is the union of vertex- h + n1 + n2 2 − disjoint pieces, Ci,i = 1, ..., z. Each piece if h 1 ≥ is either a null graph Nj, belonging either to K or to S; or matching Mr of dimension If the complete bipartite graph is the star, h of K V onto S V , r = 1,...µ; r r ⊆ K r ⊆ S The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 27 or antimatching A of dimension h of K In particular, in the special case of threshold t t t ⊆ VK onto St VS ,t =1,...α (Fig. 12.b). graphs an optimal L(2, 1)-labelling is provided b. The linear ordering⊆ C , ...C is such that with λ 2∆ + 1 (the exact values depends 1 z 2,1 ≤ each node in VK belonging to Ci is not on the graph). The optimal algorithm for linked to any node in VS belonging to Cj threshold graphs matches the polynomiality , j = 1, ..., i 1, but is linked by a black result of Chang and Kuo on cographs [42], as − edge to every node in VS belonging to Cj , threshold graphs are a subclass of cographs. j = i+1, ..., z. Furthermore, any two nodes For the more general class of unigraphs, in in VK are linked by a black edge (Fig. 12.c). [36] a 3/2-approximate algorithm for L(2, 1)- A graph is matrogenic [73] if and only if labeling this class of graphs is proposed. This its node set V can be partitioned into three algorithm runs in O(n) time, improving the disjoint sets VK , VS , and VC such that: time of the algorithm based on the greedy a. VK VS induces a split matrogenic graph technique, requiring O(m) time, that may be in which∪ K is the clique and S the stable near to Θ(n2) for unigraphs. set; Open Problem: It is still not known if b. VC induces a crown, i.e. either a perfect matching or a h-hyperoctahedron (that is the L(2, 1)-labelling problem is NP-hard the complement of a perfect matching of for unigraphs and matrogenic graphs or not. Furthermore, the cited results are dimension h – or a chordless C5; the only ones present in the literature c. every node in VC is adjacent to every node concerning these graphs, so it is probably in VK and to no node in VS. Observe that split matrogenic graphs are possible to refine the algorithm in order to improve the upper bound on λ2,1. matrogenic graphs in which VC = . A result in [73] is that a graph G∅= (V, E) is matrogenic if and only if it does not contain the 4.9.6. q-Inductive Graphs configuration in Fig. 13.a. A graph G = (V, E) Let q be a positive integer. A class of graphs is matroidal if and only if it contains neither is q-inductive if for every G , the nodes G ∈ G the configuration in Fig. 13.a nor a chordless of G can be assigned distinct integers in such C5 [149]. a way that each node is adjacent to at most q higher numbered nodes. Several well known classes of graphs belong the q-inductive class for appropriate values of q. For example, trees are 1-inductive, outerplanar graphs are 2-inductive, planar a b graphs are 5-inductive, chordal graphs with maximum clique size ω are (ω 1)-inductive FIGURE 13. The forbidden configuration of: a. and graphs of treewidth t are t-inductive.− a matrogenic graph and b. a threshold graph: — shows a present edge, - - - shows an absent edge. L(1, 1)-labelling. In [129] it is presented an approximation algorithm for L(1, 1)-labelling q-inductive graphs having performance ratio The vicinal preorder on V (G) is defined at most 2q 1. The running time of this as follows: x y iff N(x) y N(y) x, algorithm is −O(nq∆). where N(x) is the set of x’s− adjacent⊆ nodes.− A graph G is a threshold graph if and only if L(h, k)-labelling. Halld´orson [92] applies G is a split graph and the vicinal preorder on his greedy algorithm for bipartite graphs to V (G) is total, i.e. for any pair x, y V (G), q-inductive graphs, achieving a performance threshold∈ ratio of at most 2q 1, hence generalizing the either x y or y x. G is if and − only if it does not contains the configuration result for L(1, 1)-labelling to all values of h and in Fig. 13.b. k. L(2, 1)-labelling. A linear time algorithm Open Problem: Observe that for outerpla- for L(2, 1)-labelling matrogenic graphs is nar and planar graphs the bound of 2q 1 provided in [37]. Upper bounds for the is rather far from optimum (see Section− specific subclasses defined above are proved. 4.6), so probably the cited algorithm can The Computer Journal, Vol. ??, No. ??, ???? 28 T. Calamoneri KS KS C1 C2 C3 C a b 4 c FIGURE 12. a. A split matrogenic graph; b. its red graph; c. its black graph. be improved in order to guarantee a bet- for all generalized Petersen graphs of orders ter performance ratio for all values of q. 9, 10, 11 and 12 are given, thereby closing all open cases up to order N = 12 and lowering 4.9.7. Generalized Petersen Graphs the upper bound on λ2,1 down to 6 for all but For n 3, a 3-regular graph G with n = 2N three graphs of these orders. nodes≥ is a generalized Petersen graph of order Open Problem: It is not known whether N if and only if G consists of two disjoint N- there is a generalized Petersen graph of cycles, called inner and outer cycles, such that order greater than 11 with λ 7. In [5] 2,1 ≥ each node on the outer cycle is adjacent to a seven generalized Petersen graphs of order node on the inner cycle (see Figure 14). In at most 11 having λ 7 are shown. 2,1 ≥ applications involving networks, one seeks to If there is no such a graph for any order find a balance between network connectivity, greater than 11, then λ2,1 for generalized efficiency, and reliability. The double-cycle Petersen graphs of order greater than 6 structure of the generalized Petersen graphs would be at most 6, lower than the upper is appealing for such applications since it bound of 7 conjectured by Georges and is superior to a tree or cycle structure as Mauro [79]. it ensures network connectivity in case of Open Problem: Both to solve the men- any two independent node/connection failures tioned conjecture and to increase the or- while keeping the number of connections at a der N for which it is known λ2,1would be minimum level. interesting issues, although such results would have a graph theoretic flavor, more L(2, 1)-labelling. The λ2,1-number of every generalized Petersen graph is bounded than algorithmic. from above by 8, with the exception of In [4, 5] some subclasses of generalized the Petersen graph itself, having λ2,1-number Petersen graphs, particularly symmetric, are equal to 9. This bound can be improved to considered. The authors provide the exact 7 for all generalized Petersen graphs of order λ2,1-numbers of such graphs, for any order. N 6 [79]. The authors conjecture that the≤ Petersen graph is the only connected 3- 4.9.8. Comparability and Co-Comparability regular graph with λ2,1-number 9 and that Graphs there are neither generalized Petersen graphs A graph is a comparability graph if and only if nor 3-regular graphs with λ2,1-number 8, i.e. 7 there exists an order of its nodes v0 < v1 < is an upper bound also for generalized Petersen ... < vn 1 such that for each i The Computer Journal, Vol. ??, No. ??, ???? The L(h, k)-Labelling Problem 29 FIGURE 14. Some generalized Petersen graphs for all values of n from 3 to 7. The class of co-comparability graphs con- co-comparability graphs, so a complexity tains all graphs that are the complement of result for this class would imply results on a comparability graph. From the definition of its superclasses. comparability graph, if G is a co-comparability graph, then there exists an ordering of the 4.9.9. Sierpi´nski Graphs nodes set such that, if vi < vj < vl and The nodes of a Sierpi´nski graph S(n, k), n, k n≥ (vi, vl) E then either (vi, vj ) E or 1, are labelled with strings in 0,...,k 1 ; ∈ ∈ { − } (vj , vl) E. two different nodes u = (u ,...,u ) and v = ∈ 1 n Co-comparability graphs are also perfect (v1,...vn) are adjacent if and only if there graphs and include interval and permutation exists an index h such that: graphs. (i) ut = vt for t =1,...,h 1; (ii) u = v and − L(1, 1)-labelling. As the square of a h 6 h comparability graph G is G itself and (iii) ut = vh and vt = uh for t = h +1,...,n. co-comparability graphs are closed under The graphs S(2, 3) and S(3, 3) are shown in powers [51], in view of the fact that both Figure 15. comparability and co-comparability graphs are 00 00 01 10 11 perfect, it easily follows that the L(1, 1)- 02 13 labeling problem is polynomially solvable on 02 01 03 12 these classes of graphs. 20 10 30 21 31 20 L(h, k)-labelling. A co-comparability graph can be L(h, k)-labeled with span at most 22 21 12 11 00 01 10 11 max(h, 2k)2∆+ k [33]. This result is obtained exploiting the linear order of the nodes of co- FIGURE 15. S(2, 3) and S(3, 3). comparability graphs and some considerations on the degree of nodes based on the property of their edges. L(2, 1)-labelling. Gravier, Klav˘zar and Mollard [85] prove that for any n 2 and any Open Problem: Comparability and co- k 3, λ (S(n, k))=2k. ≥ comparability graphs are very interesting ≥ 2,1 graphs containing many classes, so they ACKNOWLEDGMENTS deserve to be better investigated; above all, it would be interesting to understand I am keeping a continuously updated version whether the L(2, 1)-labelling problem of this paper at the web page is still polynomially solvable or not. www.dsi.uniroma1.it/ calamo/survey.htm, so Observe that interval graphs lie in the I would appreciate to∼ receive news about new intersection between comparability and results in order to update this collection of The Computer Journal, Vol. ??, No. ??, ???? 30 T. Calamoneri references. Every note highlighting missing [8] Agnarsson, G. and Halld´orsson, M. (2003) results, possible errors and further references Coloring Powers of planar graphs. SIAM is wellcome. Journal on Discrete Mathematics, 16(4), I would like to thank the anonymous referees 654–662. that have carefully read both the first and the [9] Alon, N. and Mohar, B. (2002) The chromatic updated version of this survey and annotated number of graph powers. Combinatorics, bibliography: their comments and suggestions Probability and Computing, 11, 1–10. have been very useful. Furthermore, a special [10] Aly, K.A. and Dowd, P.W. (1994) A appreciation goes to all those people that have Class of Scalable Optical Interconnection Networks through Discrete Broadcast-Select sent me their papers either unpublished or still Multi-Domain WDM. Proceedings of IEEE in press and those that have contributed to INFOCOM, Toronto, Ontario, Canada, 12– enrich the web page associated to the paper: 16 June, pp. 392–399, IEEE Computer each of their contributions has been precious Society Press. and thanks to all of them I could collect so [11] Araki, T. 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