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

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(1)A Linear Time Algorithm for Solving the Incidence Coloring Problem of Chordal Graphs Yen-Ju Chen1 , 1. Shyue-Ming Tang2. and. Yue-Li Wan3. ∗. Department of Information Management,. National Taiwan University of Science and Technology, Taipei, Taiwan. 2. Department of Psychology,. National Defense University, Taipei, Taiwan. 3. Department 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 vertex and one of is colored with c. The incidence coloring number of its incident edge in G. The incidence coloring prob- G, denoted by χ (G), is the smallest size of the ι 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. Let ∆(G) be the maximum degree of a graph G. do not dye the same color. A graph G is called a Then, it is obvious that χι (G) ≥ ∆(G) + 1 if G has chordal (or triangulated) graph if and only if there at least one edge. Brualdi and Massey have proved is no induced cycle of length greater than 3 in G. 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 colKeywords: chordal graphs, incidence coloring problem, perfect elimination ordering.. oring problem is a special case of directed star arboricity which was introduced by Algor and Alon [1]. Meanwhile, the directed star arboricity problem has application in the WDM (Wavelength Division Multiplexing) of a star optical network [2].. 1 . Introduction. As for the incidence coloring number of special classes of graphs, the following results are wellThe incidence set of a graph G = (V, E) is defined known: as I(G) = {(v, e) : v ∈ V, e ∈ E, v is incident with • For every n ≥ 2, χι (Kn ) = n = ∆(Kn ) + 1 [3], e}, where V and E are the vertex and edge, rewhere 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 biedge v1 v2 equals to e1 or e2 . partite graph with m, n vertices in two partite sets. An incidence coloring function σ of G is a map∗ All. correspondence should be addressed to Professor Yue-Li Wang, Department of Computer Science and Information Engineering, National Chi-Nan University, 1 University Rd. Puli, Nantou, Taiwan 545. (Email: yuelwang@ncnu.edu.tw).. • For every tree T of order n ≥ 2, χι (T ) = ∆(T ) + 1 [3]. • For every Halin graph G with ∆(G) ≥ 5, χι (G) = ∆(G) + 1 [12].. - 215 -.

(2) • For every outerplanar graph G with ∆(G) ≥ 4, A vertex u ∈ N (v) is called a higher neighbor of χι (G) = ∆(G) + 1 [12]. v if ρ−1 (u) > ρ−1 (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 conjecNh (v) = {u ∈ N (v) : ρ−1 (u) > ρ−1 (v)}. ture holds for cubic Hamiltonian graphs and some 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., l Huang et al. showed that square mesh, hexagonal meshes and honeycomb meshes can be incidenceNl (v) = {u ∈ N (v) : ρ−1 (u) < ρ−1 (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 Nh (v) and Nl (v), repectively. degenerated graph G is at most ∆(G) + 2k − 1. Chordal graphs form an important and widely studied subclass of perfect graphs. Further, chordal graphs have applications in many practical areas such as scheduling, Gaussian elimination on sparse matrices, and so on [6]. In this paper, we shall propose a linear time algorithm for incidence-coloring a chordal graph G. In addition, we also prove that the incidence coloring number of a chordal graph G is ∆(G) + 1. The remaining part of this paper is organized as follows. In Section 2, we introduce chordal graphs and some important properties of chordal graphs. Section 3 contains our incidence coloring algorithm and the correctness proof of the algorithm. The last section gives our conclusion and idea for future work.. A chordal graph G can be constructed by reversing a PEO ρ. That is, starting with an empty graph, we add vertices according to the order ρ(n), ρ(n−1), . . . , ρ(1) and make each added vertex v adjacent to all vertices in S Nh (v). Let G[v] be the subgraph induced by {v} Nh (v), or Nh [v], in G. By Theorem 1, it turns out that G[v] is a clique. We can determine the incidence coloring number of G[v] by using a previous result proposed in [3]. Lemma 2 For each vertex v in a chordal graph G, χι (G[v]) = dh (v) + 1. Proof. Brualdi and Massey have proved that for every n ≥ 2, χι (Kn ) = ∆(Kn ) + 1. Since G[v] 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 assign distinct color to every vertex v in G, and call An undirected graph is chordal if and only if there it the attached color of v. Then, the attached color is no induced cycle of length greater than three. Let of vertex v is used to dye incidence (u, uv) for each N (v) denote the setSof neighbors of v and N [v] devertex u ∈ N (v). This scheme can be extended to note the set of {v} N (v). A vertex v of a graph dye incidences of a chordal graph. G = (V, E) is called simplicial if N (v) induces a clique in G. An elimination ordering ρ of a graph G The union of two graphs GS 1 = (V1 , E1 ) and 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 −1 tion ordering and ρ (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 veran 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 for a chordal graph. For example, the lexicographic breadth-first search algorithm proposed by Rose et al. is the most famous one [10]. Given a PEO ρ of a chordal graph G, we have the following definitions.. Proof. We prove the lemma by induction on the cardinality of Si . When i = n − 1, G[Sn−1 ] is a 2-clique that consists of vertices ρ(n) and ρ(n − 1) if G is connected. It is obviously true that χι (G[Sn−1 ]) = ∆(G[Sn−1 ]) + 1 = 2 since two. 2 - 216.

(3) attached colors are required for a 2-clique. (In case that G is disconnected, we can get the incidence coloring number of individual connected component and solve the problem.) Suppose χι (G[Sk+1 ]) = ∆(G[Sk+1 ]) + 1 is true. There are two conditions after ρ(k) is added to the simplicial vertex set Sk+1 . One condition is that every vertex in G[ρ(k)] has the maximum degree and no other vertex in G[Sk ] has the maximum degree. In this case, we have ∆(G[Sk ]) = ∆(G[Sk+1 ]) + 1 since the increased degree must due to the added ρ(k). Based on the coloring scheme used in complete graph, incidence (ρ(k), ρ(k)u) is dyed with the attached color of vertex u for every vertex u ∈ Nh (ρ(k)). As for the attached color of ρ(k), it is inevitable to assign a new color. This newly-assigned color is used to dye incidence (u, uρ(k)) for every vertex u ∈ Nh (ρ(k)). As a result, χι (G[Sk ]) = χι (G[Sk+1 ]) + 1 = ∆(G[Sk+1 ]) + 1 = ∆(G[Sk ]).. Step 2. Incidence coloring G. k ← 0; For i = n downto 1 do If IC[ρ(i)] is null then k ← k + 1; IC[ρ(i)] ← ck ; Endif For each u ∈ Nl (ρ(i)) do σ(u, uρ(i)) ← IC[ρ(i)]; If IC[u] is null then k ← k + 1; IC[u] ← ck ; Endif σ(ρ(i), ρ(i)u) ← IC[u]; Enddo Enddo χι (G) ← k; Step 3. Output the incidence coloring number χι (G). End of Algorithm InciColor Chordal. The other condition is that there exists a vertex w ∈ G[Sk+1 ] and w 6∈ G[ρ(k)] such that dh (ρ(k)) is less than or equal to the degree of w. That is, ∆(G[Sk+1 ]) = ∆(G[Sk ]). Since w 6∈ Nh (ρ(k)), The attached color of w can be assigned to the attached color of vertex ρ(k) and complete the incidence-coloring work. In this case, χι (G[Sk ]) = χι (G[Sk+1 ]) = ∆(G[Sk+1 ]) = ∆(G[Sk ]). . We give an example to illustrate the incidence coloring algorithm. Considering the chordal graph G shown in Figure 1(a), a PEO ρ = {v3 , v5 , v6 , v4 , v2 , v1 } of G is shown in Figure 1(b). We start with vertex v1 . Since IC[v1 ] is null, we assign a color c1 to IC[v1 ]. Three vertices v2 , v4 and v6 are adjacent to v1 , i.e., Nl (v1 ) = {v2 , v4 , v6 }. Accordingly, we get σ(v2 , v2 v1 ) = Furthermore, since G[S1 ] = G, we have the fol- σ(v4 , v4 v1 ) = σ(v6 , v6 v1 ) = c1 . Then, we assign c2 , c3 and c4 to IC[v2 ], IC[v4 ] and IC[v6 ], respeclowing corollary. tively, such that σ(v1 , v1 v2 ) = c2 , σ(v1 , v1 v4 ) = c3 , Corollary 4 For a chordal graph G, χι (G) = and σ(v1 , v1 v6 ) = c4 . As processing vertex v2 , since IC[v2 ] has been set to c2 and Nl (v2 ) = ∆(G) + 1. {v3 , v4 , v5 , v6 }, we get σ(v3 , v3 v2 ) = σ(v4 , v4 v2 ) = σ(v5 , v5 v2 ) = σ(v6 , v6 v2 ) = c2 . Then, we assign c5 and c6 to IC[v3 ] and IC[v5 ], respectively, 3 . The Incidence Coloring Al- and obtain σ(v2 , v2 v4 ) = c3 , σ(v2 , v2 v6 ) = c4 , gorithm σ(v2 , v2 v3 ) = c5 , and σ(v2 , v2 v5 ) = 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 corvertex and compute the incidence coloring number rectly incidence-color a chordal graph in O(m + n) time, where m and n are the size and order of the of the graph. 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 colProof. Algorithm InciColor Chordal is correct ored with color IC[v] in our algorithm. 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.P To color every incidence in the 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. 3 - 217.

(4) v1. c2. c1. c3. c4 c6. c4. v2. (URL:http://www.ipm.ac.ir/combinatoricsII/ abstracts/Amini.pdf). c5. c3 c2. v3 c1. v6. c1 c2. c2 c3. c4. c6. v5. c5. v4 c6. c2 c4. c3. IC[ρ(i)]. ρ(i). 6. v1. v2 v4 v6. c1. 5. v2 v4. v3 v4 v5 v6 v3 v5 v6. c2 c3. v6 v5 v3. v5. c4 c6 c5. 4 3 2 1. − −. c3. (a). (b). Figure 1: An example of Algorithm InciColor Chordal: (a) a chordal graph and its incidence coloring; (b) a PEO of the graph and the related data of incidence coloring.. [3] R. A. Brualdi and J. Q. Massey, Incidence and Strong Edge Color Graphs, Discrete Mathematics, Vol. 122, 1993, pp. 51-58. [4] M. H. Dolamaa, E. Sopenaa and X. Zhub, Incidence Coloring of k-degenerated Graphs, Discrete Mathematics, Vol. 283, 2004, pp. 121128.. 4 . Concluding Remarks We have proposed a linear time algorithm for incidence-coloring a chordal graph and proved that the incidence coloring number of a chordal graph G is ∆(G) + 1. The future research works are summarized as two directions. One is to find out other classes of graphs which have the property of χι (G) = ∆(G) + 1. Another is to find out other variations of graph coloring problem which have solutions in complete graphs, and to extend the solutions to chordal graphs.. Acknowledgement This research was supported by National Science Council under the Grants NSC95-2221-E-260-025 and NSC94-2115-M-135-001.. References. Nl (ρ(i)). i. [5] D.R. Fulkerson and O.A. Gross, Incidence Matrices and Interval Graphs, Pacific Journal of Mathematics, Vol. 15, 1965, pp. 835-855. [6] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980. [7] B. Guiduli, On Incidence Coloring and Star Arboricity of Graphs, Discrete Mathematics, Vol. 163, 1997, pp. 275-278. [8] H. I. Huang, Y. L. Wang and S. S. Chung, On the Incidence Coloring Numbers of Meshes, Computers and Mathematics with Applications, Vol. 48, 2004, pp. 1643-1649. [9] M. Maydanskiy, The Incidence Coloring Conjecture for Graphs of Maximum Degree 3, Discrete Mathematics, Vol. 292, 2005, pp. 131141. [10] D.J. Rose, R.E. Tarjan and G.S. Leuker, Algorithmic Aspects of Vertex Elimination on 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, On Incidence Coloring for Some Cubic Graphs, Graphs, Discrete Mathematics, Vol. 75, 1989, Discrete Mathematics, Vol. 252, 2002, pp. 259pp. 11-22. 266. [2] O. Amini, WDM and Directed Star Arboricity [12] S.D. Wang, D.L. Chen and S.C. Pang, The Incidence Coloring Number of Halin Graphs of Digraphs, IPM Combinatorics II: Design and Outerplanar Graphs, Discrete MathematTheory, Graph Theory, and Computational ics, Vol. 256, 2002, pp. 397-405. Methods, April 22-27, 2006, IPM, Tehran. 4 - 218.

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