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The (D + 2,2)-Incidence Coloring of Outerplanar Graphs

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The (D + 2,2)-Incidence Coloring of Outerplanar Graphs Available online at www.sciencedirect.com Progress in Natural Science 18 (2008) 575–578 www.elsevier.com/locate/pnsc The (D + 2,2)-incidence coloring of outerplanar graphs Shudong Wang a,b,c,*, Jin Xu a,b, Fangfang Ma c, Chunxiang Xu a,b a Institute of Software, School of Electronic Engineering and Computer Science, Peking University, Beijing 100871, China b Key Laboratory of High Confidence Software Technologies, Ministry of Education, Beijing 100871, China c College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266510, China Received 17 July 2007; received in revised form 24 September 2007; accepted 27 September 2007 Abstract An incidence coloring of graph G is a coloring of its incidences in which neighborly incidences are assigned different colors. In this paper, the incidence coloring of outerplanar graphs is discussed using the techniques of exchanging colors and the double inductions from the aspect of configuration property. Results show that there exists a (D + 2,2)-incidence coloring in every outerplanar graph, where D is the maximum degree of outerplanar graph. Ó 2008 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in China Press. All rights reserved. Keywords: Incidence coloring; Incidence chromatic number; Outerplanar graph 1. Introduction complete graphs and complete bipartite graphs, and put forward the incidence coloring conjecture (ICC). They sup- The concept of incidence coloring was introduced by posed that the incidence chromatic number vi(G) is always Brualdi and Massey in 1993 [1]. Let G =(V,E) be a mult- less than or equal to the maximum degree of the graph plus igraph of order p and size q, I(G)={(v,e)jv 2 V,e 2 E 2. Guiduli [2] found that the incidence coloring is a special and v is incident with e} be the set of incidences of G. case of directed star arboricity introduced by Algor and We say that two incidences (v,e) and (w,f) are neighborly Alon [3], and he also proved that ICC is incorrect. More- if one of the following conditions holds: over, according to a tight upper bound for directed star arboricity, Guiduli gave an upper bound for incidence (i) v = w; chromatic number, vi(G) 6 D + O(logD). Although ICC (ii) e = f; has been proved incorrect, how to determine the incidence (iii) the edge vw equals e or f. chromatic number of graph is fascinating. Chen et al. [4] determined the incidence chromatic numbers of paths, An incidence coloring of graph G is a coloring of its inci- cycles, fans, wheels, adding-edge wheels and complete 3- dences in which neighborly incidences are assigned different partite graphs. Then, the incidence chromatic numbers of colors. The incidence chromatic number of G, denoted by many types of graphs were determined [5–8]. Later, Shiu vi(G), is the smallest number k of colors such that there and Sun [9] gave a counter example to show that outerpla- exists a k-incidence coloring in G. Brualdi and Massey [1] nar graph of D = 4 is not 5-incidence colorable. This con- determined the incidence chromatic numbers of trees, tradicts the incidence chromatic number of outerplanar graphs proved in Ref. [10]. This contradiction results from using the following Statement 1 several times which, how- * Corresponding author. Tel.: +86 13681078897. ever, is incorrect in the proof of the incidence chromatic E-mail address: [email protected] (S. Wang). number of outerplanar graphs [10]. Thus, the proof of 1002-0071/$ - see front matter Ó 2008 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in China Press. All rights reserved. doi:10.1016/j.pnsc.2007.09.007 576 S. Wang et al. / Progress in Natural Science 18 (2008) 575–578 the incidence chromatic number of outerplanar graphs Lemma 4. [6] Graph Sp admits a (p,1)-incidence coloring. should be void and the problem of determining the inci- dence chromatic number of 2-connected outerplanar Lemma 5. [8] Let G be a graph of D(G) = 3. Then graphs is still open. In this paper, the incidence coloring vi(G) 6 5. of outerplanar graphs is discussed using the techniques of exchanging colors and the double inductions from the Lemma 6. Let G be an outerplanar graph of D(G) = 3. Then aspect of configuration property. Results show that there there exists a (5,2)-incidence coloring in G. exists a (D + 2,2)-incidence coloring in every outerplanar graph, where D is the maximum degree of outerplanar Proof. According to Lemma 5, for "v 2 V(G), there are at graph. most two colors in Av. Hence, the above lemma holds. h Statement 1. [11] Let v be a cut-vertex of outerplanar graph G, G À v = H1 [ H2, G1 = G[V(H1) [ {v}], G2 = Lemma 7. [12] Let G be a connected graph. Then G[V(H2) [ {v}]. Then q(G) P p(G) À 1. viðGÞ¼maxfviðG1Þ; viðG2Þ; dGðvÞþ1g: Lemma 8. Let G be a 2-connected graph. Then q(G) P 2D(G) À 1. 2. Lemmas and notations Proof. Let v be a maximum degree-vertex of G. Let * * In this study, we always limit to finite, simple, and undi- G = G À v. By the 2-connectivity of G, G is also con- * * rected graphs. In a given graph G, the vertex of degree k is nected. According to Lemma 7, q(G ) P p(G ) À 1. Because q(G*)=q(G) À D(G), p(G*)=p(G) À 1, q(G)= called k-vertex. p(G), q(G), D(G), and NG(v) denote the * * order, the size, the maximum degree and the set of vertices q(G )+D(G) P p(G ) À 1+D(G)=p(G) À 2+D(G). Again adjacent to v of G, respectively, and p, q, D, N(v) for short, since G is a simple graph, p(G) À 1 P D(G). Therefore, h respectively. For simplicity, we denote the set of all the q(G) P 2D(G) À 1. incidences of the form (u,uv) and (v,vu)ofG by Av(G) Lemma 9. [13] Let G be a 2-connected outerplanar graph of and Iv(G) (abbreviated as Av and Iv), respectively, where u is an adjacent-vertex of v. Let r be a k-incidence coloring order p(G) P 3. Then one of the following conditions holds: of G. We denote by r(uv) the ordered pair (r(u,uv), r(v,vu)) of two colors r(u,uv), r(v,vu), and Fr(u,uv) the unavailable (i) There exist two neighborly 2-vertices u and v. color set for the incidence (u,uv)inr, namely, Fr(u,uv)= (ii) There exists a 2-vertex u adjacent to a 3-vertex v. Let N(u) = {v,u1},vu1 2 E(G). r(Iu) [ r(Au) [ r(Iv). In a k-incidence coloring r of G,if (iii) There exists a 2-vertex u adjacent to two neighborly 4- for "v 2 V(G), jr(Av)j 6 l, then we call r a(k,l)-incidence coloring of G. The terms and notations not stated here vertices v and w. Let N(v) = {u,w,v1,v2},N(w) = can be found in Ref. [12]. {u,v,w1,w2}, and d(v1)=d(w1) = 2,v1v2,w1w2 2 E(G). In addition, we need the following lemmas to obtain the (iv) There exist two non-neighborly 2-vertices u and v adja- main results. cent to a 4-vertex w. Let N(w) = {u,v,w1,w2},uw1, vw2,w1w2 2 E(G). Lemma 1. Each 2-connected graph is 2-edge-connected. 3. Main results and proofs Proof. (disproof) Suppose that G is 2-connected but not 2- edge-connected. Then there exists an edge e = uv in G such In the following proofs, we always assume that the that G À e is disconnected. Thus G À u is also discon- graphs are 2-connected. nected. That is to say, u is a cut-vertex of G. This contra- dicts that G is 2-connected graph. The proof is Theorem 1. Let G be a graph of D(G) = D P 4, completed. h q(G) = 2D À 1. Then G admits a (D + 2,2)-incidence coloring. Let G =(V,E) be a planar graph. If $v 2 V(G) such that G À v is a forest, then we call G 1-tree graph. We denote by Proof. Let v be a maximum degree-vertex of G. Since G is * Sp the graph with p vertices u,v,x1,x2,ÁÁÁ,xpÀ2 and 2p À 4 2-connected, G = G À {v} is also connected. According to edges ux1,ux2,ÁÁÁuxpÀ2,vx1,vx2,ÁÁÁ,vxpÀ2. the known condition that q(G)=2D À 1, we obtain that D À 1=q(G) À D = q(G*) P p(G*) À 1=p(G) À 2, and so Lemma 2. [6] Let G be a 2-edge-connected 1-tree graph of D P p(G) À 1. Based on the fact that D 6 p(G) À 1, we * * D(G) = D P 4 and G 6¼ Sp. Then vi(G) = D +1. obtain D = p(G) À 1. Therefore, q(G )=p(G ) À 1. By the According to Lemmas 1 and 2, we may obtain the definition of tree, G* is a tree.
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