A Survey on the Distance-Colouring of Graphs Florica Kramer, Horst Kramer

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Discrete Mathematics 308 (2008) 422–426 www.elsevier.com/locate/disc

A survey on the distance-colouring of graphs Florica Kramer, Horst Kramer

Nesselweg 39, D 65527 Niedernhausen, Germany

Received 22 November 2004; received in revised form 25 October 2005; accepted 27 November 2006 Available online 2 June 2007

Abstract A distance-colouring relative to distance p of a graph G = (V, E) is a mapping of V in a set of colours in such a way that any two vertices of G of distance not greater than p have distinct colours. The purpose of this paper is to give a survey of the results related to the distance-colouring of graphs introduced in 1969. © 2007 Elsevier B.V. All rights reserved.

Keywords: Distance-colouring; Planar graphs; Lattices

1. Introduction

In 1969 we defined [22,23] the chromatic number (p, G) relative to distance p of a graph G = (V, E) to be the minimum number of colours which are sufficient for colouring the vertices of G in such a way that any two vertices p of G of distance not greater than p have distinct colours. For a natural number p we define the power G = (V, Ep) as the graph with the same vertex set and with an edge between any pair of vertices that are distance at most p away from each other in G. We have then obviously (p, G) = (Gp), where (Gp) is the usual chromatic number of the pth power of G. In the seventies the problem was studied by Kramer [21], Speranza [29], Wegner [31] and Antonucci [1].Inthe eighties the same topic was known as Ls-colouring especially in the Italian literature (the papers of Gionfriddo [7–13], Gionfriddo and Milici [14], Gionfriddo et al. [15] and others). In the nineties the problem became known as the distance-colouring problem (see the book of Jensen and Toft [20, Section 2.18] and [16,33]). In the following we shall consider only undirected, finite, connected graphs without loops and multiple edges. The maximum vertex degree of G is denoted in the following by . We shall denote by r the floor and by r the ceil of a real number r. The distance d(u, v) between two vertices u and v in a connected graph G = (V, E) is the length of a shortest path (often called a geodesic) joining them. The diameter D(G) of a connected graph G is defined to be the length of any longest geodesic. The distance between two subsets V1,V2 of the vertex set V of a connected graph G = (V, E) is defined by d(V1,V2) = min{d(x,y); x ∈ V1,y∈ V2}. The diameter D(X) of a subset X ⊆ V of the vertex set V of a connected graph G = (V, E) is D(X) = max{d(x,y); x ∈ X, y ∈ X}.

E-mail address: [email protected] (F. Kramer).

2. Results

We gave in [22,23] the following characterization of the graphs for which we have (p, G) = p + 1 (cited in [2]). The same result was found in 1975 by Speranza [29].

Theorem 1. ForagraphG = (V, E) we have (p, G) = p + 1 if and only if the graph G is satisfying one of the following conditions:

(a) | V |=p + 1, (b) G is a path of length greater than p, (c) G is a cycle of length a multiple of (p + 1).

The ﬁrst reference on the colouring squares of planar graphs was the unpublished paper [31] of Wegner. In this paper, Wegner conjectured that:

Conjecture. Let G be a planar graph. Then

(2,G) + 5if47 and

(2,G)3/2+1if8.

He proved in [31] among other results that (2,G)8 for any simple planar graph with 3. A still open problem as shown by Gionfriddo in [13] (see also [33]) is that to ﬁnd out if there are planar graphs with = 3 and (2,G)= 8ortoprove:

Open Problem. If G is a planar graph with = 3, then is (2,G)7?

Borodin [3] has solved this problem for cubic planar graphs without faces of size greater than 5 (see [20]). Some authors (among them Molloy and Salavatipour [27], Lih et al. [25]) announced in their papers, that very recently Thomassen [30] has solved this conjecture for = 3, but C. Thomassen conﬁrmed me in a private communication that the problem must be considered further on as an open problem. We found in [24] similar bounds for (3,G)for planar bipartite graphs with 3:

Theorem 2. Let G = (A, B; E) be a planar bipartite graph with 3. Then we have (3,G)8.

This upper bound cannot be improved, i.e. there are cubic planar bipartite graphs for which (3,G)= 8 (for instance the hexahedron). If the hypothesis of planarity is omitted in Theorem 2 we obtain [24]:

Theorem 3. Let G be a bipartite graph with 3. Then we have (3,G)14.

We have (3,G)= 14 for the well-known Heawood graph, a cubic graph of diameter 3 with 14 vertices. A direct generalization of Theorem 2 is the following:

Theorem 4. The chromatic number (3,G)of a bipartite graph has the upper bound:

(3,G)2(1 + ( − 1)). (1)

In [24] there are given also examples of graphs for which we have equality in (1) for the cases = 2, = 3 and = 4. The following upper bound for the distant chromatic number of planar graphs was found by Jendrol and Skupie´n [19]. 424 F. Kramer, H. Kramer / Discrete Mathematics 308 (2008) 422–426

Theorem 5. Given a planar graph G, let M = max{8, }. Then we have

3M + 3 p− (p, G)6 + ((M − 1) 1 − 1). M − 2

Antonucci [1] obtained in 1978 a lower bound for the chromatic number (2,G) as a function of the number of vertices and the number of edges of the graph.

Theorem 6. Let G = (V, E) be a simple graph with n vertices and m edges without cycles of length 3 and 4. Then we have

(2,G)n3/(n3 − 4m2). (2)

In [24] we have proved that the only graphs of diameter D(G)2 , with n vertices and m edges without cycles of length 3 and 4 for which we have the equality in (2) are the complete graphs K1,K2 and the Moore graphs of diameter 2. For completeness we shall remind the deﬁnition of the Moore graphs. Let G be a regular graph of order k with n vertices and diameter D. We have then the inequality: (D− ) n1 + k(1 + (k − 1) + (k − 1)2 +···+(k − 1) 1 ). (3)

A k-regular graph with n vertices and diameter D for which we have equality in (3) is called a Moore graph of type (k, D). In [24] we obtained a lower bound similar to the bound (2) found by Antonucci also for graphs which have cycles of length 3 and 4. The s-density of a graph G = (V, E) is deﬁned by

ds(G) = max{| X |: X ⊆ V,diam X s}.

It is then obvious that we have the inequality:

(s, G)ds(G). (4)

Let then be s(G) = (s, G) − ds(G). Gionfriddo studied the connection between the s-density of a graph ds(G) and the chromatic number (s, G) proving among others that:

Theorem 7 (Gionfriddo [7]). ∀s 3 and ∀r ∈ N ∃G (planar) such that: (s, G)ds(G) + r.

Theorem 8 (Gionfriddo [11]). For every positive integer h there exists a graph Gh such that 2(Gh) = h.

A class of graphs with D(G)>pfor which we have equality in (4) was found by Gionfriddo [9]:

Theorem 9. Let G = (V, E) be a simple graph such that ds(G)(| V |)/2.If there is a subset X of V such that | X |=ds(G), D(GX)s, D(GV −X)s then we have (s, G) = ds(G).

A generalization of this theorem is:

Theorem 10. Let G = (V, E) be a simple graph. If there exists a partition P ={X0,X1,...,Xr } of V such that:

(i) | X0 |=ds(G), (ii) D(Gi)s ∀i = 0, 1, 2,...,r, (iii) d(Xi,Xj )s + 1 ∀i, j = 1, 2,...,r, i = j, where Gi is the subgraph of G generated by the vertex set Xi, then we have

(s, G) = ds(G). F. Kramer, H. Kramer / Discrete Mathematics 308 (2008) 422–426 425

Distance-colouring was also a subject of the doctor thesis of Dillon [4] and that of Wojciechowski [32] and of the paper of Jagger [18]. Going out from the above conjecture of Wegner there have been obtained upper bounds for (2,G)by a series of authors. Wong [33] found in 1996 the following linear upper bound in terms of the maximum degree:

Theorem 11. If G is a planar graph with maximum degree then (2,G)3 + 5.

Similar linear bounds were also obtained by Wong [33] for toroidal graphs. This bound was improved by van den Heuvel and McGuinnes [16] for 20.

Theorem 12. If G is a planar graph with maximum degree then (2,G)2 + 25.

For large values of there have been found better bounds by the authors Molloy and Salavatipour [27] proving the following:

G = (V, E) ( ,G) 5 + Theorem 13. Let be a planar graph. Then 2 3 78.

G = (V, E) ( ,G) 5 + Theorem 14. For a planar graph , if 241, then 2 3 25.

13 ∗ + (3/2) The authors of the paper [6] proved the upper bound 7 O for the chromatic number of the square of a planar graph with maximum degree . A graph H is called a minor of a graph G if H can be obtained from a subgraph of G by contracting edges and G is called H -minor free if G does not have H as a minor. Lih et al. [25] obtained for K4-minor free graphs the following bounds:

Theorem 15. Let G = (V, E) be a K4-minor free graph. Then

(2,G)(G) + 3 if 2(G)3 and

(2,G)3(G)/2+1 if (G)4.

A large number of papers cited in [17,26] have found a nice application of the distance-colouring of graphs in the frequency assignment problem (radio channel assignment). The problem of frequency assignment arises when different radio transmitters are operating in a limited area. Transmitters which are too close will interfere with each other when they have the same or closely related frequency channels. The problem of assigning frequencies to the different transmitters can then be reduced to a graph colouring problem. Several authors have now studied the problem when the transmitters are spread regularly over the plane. The corresponding graph is then a lattice. Fertin et al. [5] have obtained for the square lattice:

Theorem 16. For the d-distant chromatic number (d, S) of the square lattice S we have

(d, S) =1 (d + )2 2 1 .

For the triangular lattice Sevcikova [28] proved the following:

Theorem 17. For the d-distant chromatic number (d, T ) of the triangular lattice T we have

(d, T ) =3 (d + )2 4 1 .

A similar result for the hexagonal lattice was obtained by Jacko and Jendrol [17]: 426 F. Kramer, H. Kramer / Discrete Mathematics 308 (2008) 422–426

Theorem 18. For the d-distant chromatic number (d, H ) of the hexagonal lattice H we have (d, H ) =3 (d + )2 8 1 ifdisodd and (d, H ) =3 (d + 4 )2 8 3 ifdiseven.

In conclusion we can say that the conjecture of Wegner, a simple formulated distance-colouring problem for planar graphs has resisted more than 25 years and that distance-colouring theory is also of interest for its applications.

References

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