A Linear Time Algorithm for Solving the Incidence Coloring Problem of Chordal Graphs

Yen-Ju Chen1, Shyue-Ming Tang2 and Yue-Li Wan3 ∗

1Department of Information Management, National Taiwan University of Science and Technology, Taipei, Taiwan. 2Department of Psychology, National Defense University, Taipei, Taiwan. 3Department of Computer Science and Information Engineering, National Chi-Nan University, Nantou, Taiwan.

Abstract ping from I(G) to a color set such that adjacent incidences of G are assigned different colors. For example, σ(v, e) = c means that the incidence (v, e) An incidence of G consists of a and one of is colored with c. The incidence coloring number of its incident edge in G. The incidence coloring prob- G, denoted by χι(G), is the smallest size of the lem is a variation of vertex coloring problem. The color set. The incidence coloring problem is to find problem is to find the minimum number (called in- the incidence coloring number of a given graph. In cidence coloring number) of colors needed to dye [3], Brualdi and Massey first defined the problem every incidence of G so that the adjacent incidences as a variation of vertex coloring problem. do not dye the same color. A graph G is called a Let ∆(G) be the maximum of a graph G. chordal (or triangulated) graph if and only if there Then, it is obvious that χι(G) ≥ ∆(G) + 1 if G has is no induced cycle of length greater than 3 in G. at least one edge. Brualdi and Massey have proved that the incidence coloring number of a given graph In this paper, we propose a linear time algorithm G is at most 2∆(G) [3]. They also conjectured that for incidence-coloring a chordal graph. Further, any graph G can be incidence-colored with ∆(G)+2 we prove that the incidence coloring number of a colors. However, their conjecture was disproved by chordal graph is ∆(G) + 1, where ∆(G) is the max- Guiduli [7]. imum degree of G. In [7], Guiduli also showed that the incidence col- oring problem is a special case of directed star ar- Keywords: chordal graphs, incidence coloring prob- boricity which was introduced by Algor and Alon lem, perfect elimination ordering. [1]. Meanwhile, the directed star arboricity prob- lem has application in the WDM (Wavelength Di- vision Multiplexing) of a star optical network [2]. 1 . Introduction As for the incidence coloring number of special classes of graphs, the following results are well- The incidence set of a graph G = (V,E) is defined known: as I(G) = {(v, e): v ∈ V, e ∈ E, v is incident with e}, where V and E are the vertex and edge, re- • For every n ≥ 2, χι(Kn) = n = ∆(Kn) + 1 [3], where Kn is a complete graph with n vertices. spectively, sets of G. Two incidences (v1, e1) and

(v2, e2) are adjacent if one of the following condi- • For every m ≥ n ≥ 2, χι(Km,n) = m + 2 = tions holds: (i) v1 = v2, (ii) e1 = e2, or (iii) the ∆(Km,n) + 2 [3], where Km,n is a complete bi- edge v1v2 equals to e1 or e2. partite graph with m, n vertices in two partite An incidence coloring function σ of G is a map- sets. • For every T of order n ≥ 2, χ (T ) = ∗All correspondence should be addressed to Professor ι Yue-Li Wang, Department of Computer Science and Infor- ∆(T ) + 1 [3]. mation Engineering, National Chi-Nan University, 1 Uni- versity Rd. Puli, Nantou, Taiwan 545. (Email: yuel- • For every G with ∆(G) ≥ 5, [email protected]). χι(G) = ∆(G) + 1 [12].

- 215 - • For every G with ∆(G) ≥ 4, A vertex u ∈ N(v) is called a higher neighbor of −1 −1 χι(G) = ∆(G) + 1 [12]. v if ρ (u) > ρ (v). The set of higher neighbors of v will be denoted by Nh(v), i.e.,

In [11], Shiu et al. showed that Brualdi’s conjec- −1 −1 ture holds for cubic Hamiltonian graphs and some Nh(v) = {u ∈ N(v): ρ (u) > ρ (v)}. other cubic graphs. In [9], Maydanskiy proved that Similarly, we define the set of lower neighbors of v χι(G) ≤ 5 for any graph with ∆(G) = 3. In [8], and denote it by N (v), i.e., Huang et al. showed that square mesh, hexagonal l −1 −1 meshes and honeycomb meshes can be incidence- Nl(v) = {u ∈ N(v): ρ (u) < ρ (v)}. colored with ∆(G) + 1 colors [8]. In [4], Dolama et al. proved that incidence coloring of every k- In addition, let dh(v) and dl(v) denote the size of degenerated graph G is at most ∆(G) + 2k − 1. Nh(v) and Nl(v), repectively. Chordal graphs form an important and widely A chordal graph G can be constructed by revers- studied subclass of perfect graphs. Further, chordal ing a PEO ρ. That is, starting with an empty graphs have applications in many practical areas graph, we add vertices according to the order such as scheduling, Gaussian elimination on sparse ρ(n), ρ(n−1), . . . , ρ(1) and make each added vertex matrices, and so on [6]. In this paper, we shall pro- v adjacent to all vertices in Nh(v). Let G[v] be the S pose a linear time algorithm for incidence-coloring subgraph induced by {v} Nh(v), or Nh[v], in G. a chordal graph G. In addition, we also prove that By Theorem 1, it turns out that G[v] is a clique. the incidence coloring number of a chordal graph G We can determine the incidence coloring number of is ∆(G) + 1. G[v] by using a previous result proposed in [3]. The remaining part of this paper is organized as follows. In Section 2, we introduce chordal graphs Lemma 2 For each vertex v in a chordal graph G, and some important properties of chordal graphs. χι(G[v]) = dh(v) + 1. Section 3 contains our incidence coloring algorithm and the correctness proof of the algorithm. The Proof. Brualdi and Massey have proved that for last section gives our conclusion and idea for future every n ≥ 2, χι(Kn) = ∆(Kn) + 1. Since G[v] work. is a complete subgraph induced by Nh[v] in G, the incidence coloring number of G[v] must be dh(v)+1.  2 . Preliminaries For incidence-coloring a complete graph G, we An undirected graph is chordal if and only if there assign distinct color to every vertex v in G, and call is no induced cycle of length greater than three. Let it the attached color of v. Then, the attached color N(v) denote the set of neighbors of v and N[v] de- of vertex v is used to dye incidence (u, uv) for each note the set of {v} S N(v). A vertex v of a graph vertex u ∈ N(v). This scheme can be extended to G = (V,E) is called simplicial if N(v) induces a dye incidences of a chordal graph. clique in G. An elimination ordering ρ of a graph G The union of two graphs G1 = (V1,E1) and S is a bijection ρ : {1, 2, . . . , n} → V , where n = |V |. G2 = (V2,E2), denoted by G1 G2, is the graph S S Accordingly, ρ(i) is the i-th vertex in the elimina- with vertex set V1 V2 and edge set E1 E2. We tion ordering and ρ−1(v), v ∈ V , gives the position consider the union of two subgraphs of a chordal of v in ρ.A perfect elimination ordering (PEO) is graph G. For 1 ≤ i ≤ n − 1, let Si be the ver- an elimination ordering ρ = (v1, v2, . . . , vn), where tex set {ρ(n), ρ(n − 1), . . . , ρ(i)} and G[Si] be the vi (1 ≤ i ≤ n) is a simplicial vertex in the sub- subgraph induced by Si. Then, we have G[Si] = S graph induced by vertex set {vi, vi+1, . . . , vn}. The G[Si+1] G[ρ(i)]. following theorem is well-known.

Theorem 3 Let Si = {ρ(n), ρ(n − 1), . . . , ρ(i)} be Theorem 1 (Fulkerson and Gross [5]; Golumbic a vertex set corresponding to a PEO ρ of a chordal [6]) An undirected graph is chordal if and only if it graph G. Then, χι(G[Si]) = ∆(G[Si]) + 1 for 1 ≤ has a perfect elimination ordering. i ≤ n − 1, where n is the number of vertices in G.

There exists many algorithms to generate PEOs Proof. We prove the lemma by induction on for a chordal graph. For example, the lexicographic the cardinality of Si. When i = n − 1, G[Sn−1] breadth-first search algorithm proposed by Rose et is a 2-clique that consists of vertices ρ(n) and al. is the most famous one [10]. Given a PEO ρ of a ρ(n − 1) if G is connected. It is obviously true chordal graph G, we have the following definitions. that χι(G[Sn−1]) = ∆(G[Sn−1]) + 1 = 2 since two

- 2162 - attached colors are required for a 2-clique. (In case Step 2. Incidence coloring G. that G is disconnected, we can get the incidence k ← 0; coloring number of individual connected component For i = n downto 1 do and solve the problem.) If IC[ρ(i)] is null then k ← k + 1; Suppose χ (G[S ]) = ∆(G[S ]) + 1 is true. ι k+1 k+1 IC[ρ(i)] ← c ; There are two conditions after ρ(k) is added to the k Endif simplicial vertex set S . One condition is that ev- k+1 For each u ∈ N (ρ(i)) do ery vertex in G[ρ(k)] has the maximum degree and l σ(u, uρ(i)) ← IC[ρ(i)]; no other vertex in G[S ] has the maximum degree. k If IC[u] is null then In this case, we have ∆(G[S ]) = ∆(G[S ]) + 1 k k+1 k ← k + 1; since the increased degree must due to the added IC[u] ← c ; ρ(k). Based on the coloring scheme used in com- k Endif plete graph, incidence (ρ(k), ρ(k)u) is dyed with σ(ρ(i), ρ(i)u) ← IC[u]; the attached color of vertex u for every vertex u ∈ Enddo N (ρ(k)). As for the attached color of ρ(k), it is in- h Enddo evitable to assign a new color. This newly-assigned χ (G) ← k; color is used to dye incidence (u, uρ(k)) for every ι Step 3. Output the incidence coloring number χ (G). vertex u ∈ N (ρ(k)). As a result, χ (G[S ]) = ι h ι k End of Algorithm InciColor Chordal χι(G[Sk+1]) + 1 = ∆(G[Sk+1]) + 1 = ∆(G[Sk]). The other condition is that there exists a vertex w ∈ G[Sk+1] and w 6∈ G[ρ(k)] such that dh(ρ(k)) We give an example to illustrate the inci- is less than or equal to the degree of w. That is, dence coloring algorithm. Considering the chordal ∆(G[Sk+1]) = ∆(G[Sk]). Since w 6∈ Nh(ρ(k)), graph G shown in Figure 1(a), a PEO ρ = The attached color of w can be assigned to the {v3, v5, v6, v4, v2, v1} of G is shown in Figure 1(b). attached color of vertex ρ(k) and complete the We start with vertex v1. Since IC[v1] is null, incidence-coloring work. In this case, χι(G[Sk]) = we assign a color c1 to IC[v1]. Three vertices χι(G[Sk+1]) = ∆(G[Sk+1]) = ∆(G[Sk]).  v2, v4 and v6 are adjacent to v1, i.e., Nl(v1) = {v2, v4, v6}. Accordingly, we get σ(v2, v2v1) = Furthermore, since G[S1] = G, we have the fol- σ(v4, v4v1) = σ(v6, v6v1) = c1. Then, we assign lowing corollary. c2, c3 and c4 to IC[v2], IC[v4] and IC[v6], respec- tively, such that σ(v1, v1v2) = c2, σ(v1, v1v4) = c3,

Corollary 4 For a chordal graph G, χι(G) = and σ(v1, v1v6) = c4. As processing vertex v2, ∆(G) + 1. since IC[v2] has been set to c2 and Nl(v2) = {v3, v4, v5, v6}, we get σ(v3, v3v2) = σ(v4, v4v2) = σ(v5, v5v2) = σ(v6, v6v2) = c2. Then, we as- 3 . The Incidence Coloring Al- sign c5 and c6 to IC[v3] and IC[v5], respectively, and obtain σ(v2, v2v4) = c3, σ(v2, v2v6) = c4, gorithm σ(v2, v2v3) = c5, and σ(v2, v2v5) = c6. Subsequent vertices v4, v6, v5 and v3 are processed in the same In this section, we present a linear time algorithm manner. Finally, we obtain χι(G) = 6. for incidence coloring a chordal graph. At first, we determine a PEO of the chordal graph. Based on the reversed order of the PEO, we process each Theorem 5 Algorithm InciColor Chordal can cor- vertex and compute the incidence coloring number rectly incidence-color a chordal graph in O(m + n) of the graph. time, where m and n are the size and order of the graph, respectively. Let IC[vi](i = 1, . . . , n) be an array that records the attached colors of corresponding vertices. All incidences adjacent to a common vertex v are col- ored with color IC[v] in our algorithm. Proof. Algorithm InciColor Chordal is correct since it is an implementation of the constructive We show the algorithm for incidence-coloring a proof of Theorem 3. The algorithm computes chordal graph as follows. the incidence coloring number of a chordal graph G(V,E) based on a PEO that can be obtained in Algorithm InciColor Chordal O(m + n) time. To color every incidence in the P Input: A chordal graph G. graph, it takes v∈V 2dl(v) = 2m time. Thus, the Output: Incidence coloring number χι(G). overall time requirement is O(m + n).  Step 1. Find a PEO ρ in G.

- 2173 - (URL:http://www.ipm.ac.ir/combinatoricsII/ v1 v2 abstracts/Amini.pdf)

i ρ(i) Nl(ρ(i)) IC[ρ(i)] 6 v1 c1 v3 5 v2 c2

4 v4 c3

3 c4 v6 v4 2 v5 − c6

1 v3 c5 v5

(a) (b)

Figure 1: An example of Algorithm Inci- Color Chordal: (a) a chordal graph and its inci- [3] R. A. Brualdi and J. Q. Massey, Incidence and dence coloring; (b) a PEO of the graph and the Strong Edge Color Graphs, Discrete Mathe- related data of incidence coloring. matics, Vol. 122, 1993, pp. 51-58. [4] M. H. Dolamaa, E. Sopenaa and X. Zhub, Inci- dence Coloring of k-degenerated Graphs, Dis- 4 . Concluding Remarks crete Mathematics, Vol. 283, 2004, pp. 121- 128. We have proposed a linear time algorithm for [5] D.R. Fulkerson and O.A. Gross, Incidence Ma- incidence-coloring a chordal graph and proved that trices and Interval Graphs, Pacific Journal of the incidence coloring number of a chordal graph Mathematics, Vol. 15, 1965, pp. 835-855. G is ∆(G) + 1. The future research works are [6] M.C. Golumbic, Algorithmic summarized as two directions. One is to find out and Perfect Graphs, Academic Press, New other classes of graphs which have the property of York, 1980. χι(G) = ∆(G) + 1. Another is to find out other [7] B. Guiduli, On Incidence Coloring and Star variations of problem which have so- Arboricity of Graphs, Discrete Mathematics, lutions in complete graphs, and to extend the solu- Vol. 163, 1997, pp. 275-278. tions to chordal graphs. [8] H. I. Huang, Y. L. Wang and S. S. Chung, On the Incidence Coloring Numbers of Meshes, Computers and Mathematics with Applica- Acknowledgement tions, Vol. 48, 2004, pp. 1643-1649. This research was supported by National Science [9] M. Maydanskiy, The Incidence Coloring Con- Council under the Grants NSC95-2221-E-260-025 jecture for Graphs of Maximum Degree 3, Dis- and NSC94-2115-M-135-001. crete Mathematics, Vol. 292, 2005, pp. 131- 141. [10] D.J. Rose, R.E. Tarjan and G.S. Leuker, Al- gorithmic Aspects of Vertex Elimination on References Graphs, SIAM Journal on Computing, Vol 5, 1976, pp. 266-283. [1] I. Algor and N. Alon, The Star Arboricity of [11] W. C. Shiu, P. C. B. Lam and D. L. Chen, Graphs, Discrete Mathematics, Vol. 75, 1989, On Incidence Coloring for Some Cubic Graphs, pp. 11-22. Discrete Mathematics, Vol. 252, 2002, pp. 259- 266. [2] O. Amini, WDM and Directed Star Arboricity [12] S.D. Wang, D.L. Chen and S.C. Pang, The of Digraphs, IPM Combinatorics II: Design Incidence Coloring Number of Halin Graphs and Outerplanar Graphs, Discrete Mathemat- Theory, Graph Theory, and Computational ics, Vol. 256, 2002, pp. 397-405. Methods, April 22-27, 2006, IPM, Tehran.

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