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The (Δ + 2,2)-incidence coloring of outerplanar graphs
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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. 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 The concept of incidence coloring was introduced by Brualdi and Massey in 1993 [1]. Let G = (V, E) be a multigraph of order p and size q, I(G) = {(v, e)jv 2 V,e 2 E and v is incident with e} be the set of incidences of G. We say that two incidences (v, e) and (w, f) are neighborly if one of the following conditions holds: (i) v = w; (ii) e = f; (iii) the edge vw equals e or f. An incidence coloring of graph G is a coloring of its incidences in which neighborly incidences are assigned different colors. The incidence chromatic number of G, denoted by vi(G), is the smallest number k of colors such that there exists a k-incidence coloring in G. Brualdi and Massey [1] determined the incidence chromatic numbers of trees,
*
Corresponding author. Tel.: +86 13681078897. E-mail address: [email protected] (S. Wang).
complete graphs and complete bipartite graphs, and put forward the incidence coloring conjecture (ICC). They supposed that the incidence chromatic number vi(G) is always less than or equal to the maximum degree of the graph plus 2. Guiduli [2] found that the incidence coloring is a special case of directed star arboricity introduced by Algor and Alon [3], and he also proved that ICC is incorrect. Moreover, according to a tight upper bound for directed star arboricity, Guiduli gave an upper bound for incidence chromatic number, vi(G) 6 D + O(log D). Although ICC has been proved incorrect, how to determine the incidence chromatic number of graph is fascinating. Chen et al. [4] determined the incidence chromatic numbers of paths, cycles, fans, wheels, adding-edge wheels and complete 3partite graphs. Then, the incidence chromatic numbers of many types of graphs were determined [5–8]. Later, Shiu and Sun [9] gave a counter example to show that outerplanar graph of D = 4 is not 5-incidence colorable. This contradicts the incidence chromatic number of outerplanar graphs proved in Ref. [10]. This contradiction results from using the following Statement 1 several times which, however, is incorrect in the proof of the incidence chromatic number of outerplanar graphs [10]. Thus, the proof of
1002-0071/$ - see front matter 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 should be void and the problem of determining the incidence chromatic number of 2-connected outerplanar graphs is still open. 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. Statement 1. [11] Let v be a cut-vertex of outerplanar graph G, G v = H1 [ H2, G1 = G[V(H1) [ {v}], G2 = G[V(H2) [ {v}]. Then
Lemma 4. [6] Graph S p admits a (p, 1)-incidence coloring.
vi ðGÞ ¼ maxfvi ðG1 Þ; vi ðG2 Þ; d G ðvÞ þ 1g:
Lemma 8. Let G be a 2-connected graph. Then q(G) P 2D(G) 1.
2. Lemmas and notations In this study, we always limit to finite, simple, and undirected graphs. In a given graph G, the vertex of degree k is 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 adjacent to v of G, respectively, and p, q, D, N(v) for short, respectively. For simplicity, we denote the set of all the incidences of the form (u, uv) and (v, vu) of G by Av(G) and Iv(G) (abbreviated as Av and Iv), respectively, where u is an adjacent-vertex of v. Let r be a k-incidence coloring 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 color set for the incidence (u, uv) in r, namely, Fr(u, uv) = r(Iu) [ r(Au) [ r(Iv). In a k-incidence coloring r of G, if 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 can be found in Ref. [12]. In addition, we need the following lemmas to obtain the main results. Lemma 1. Each 2-connected graph is 2-edge-connected. Proof. (disproof) Suppose that G is 2-connected but not 2edge-connected. Then there exists an edge e = uv in G such that G e is disconnected. Thus G u is also disconnected. That is to say, u is a cut-vertex of G. This contradicts that G is 2-connected graph. The proof is completed. h Let G = (V, E) be a planar graph. If $v 2 V(G) such that G v is a forest, then we call G 1-tree graph. We denote by S p the graph with p vertices u, v, x1, x2, , xp2 and 2p 4 edges ux1, ux2, uxp2, vx1, vx2, , vxp2. Lemma 2. [6] Let G be a 2-edge-connected 1-tree graph of D(G) = D P 4 and G 6¼ S p . Then vi(G) = D + 1. According to Lemmas 1 and 2, we may obtain the following Lemma 3. Lemma 3. Let G be a 2-connected 1-tree graph of D(G) = D P 4 and G 6¼ S p . Then vi(G) = D + 1.
Lemma 5. [8] Let G be a graph of D(G) = 3. Then vi(G) 6 5. Lemma 6. Let G be an outerplanar graph of D(G) = 3. Then there exists a (5, 2)-incidence coloring in G. Proof. According to Lemma 5, for "v 2 V(G), there are at most two colors in Av. Hence, the above lemma holds. h Lemma 7. [12] Let G be a connected graph. Then q(G) P p(G) 1.
Proof. Let v be a maximum degree-vertex of G. Let G* = G v. By the 2-connectivity of G, G* is also connected. According to Lemma 7, q(G*) P p(G*) 1. Because q(G*) = q(G) D(G), p(G*) = p(G) 1, q(G) = q(G*) + D(G) P p(G*) 1 + D(G) = p(G) 2 + D(G). Again since G is a simple graph, p(G) 1 P D(G). Therefore, q(G) P 2D(G) 1. h Lemma 9. [13] Let G be a 2-connected outerplanar graph of order p(G) P 3. Then one of the following conditions holds: (i) There exist two neighborly 2-vertices u and v. (ii) There exists a 2-vertex u adjacent to a 3-vertex v. Let N(u) = {v, u1}, vu1 2 E(G). (iii) There exists a 2-vertex u adjacent to two neighborly 4vertices v and w. Let N(v) = {u, w, v1, v2},N(w) = {u, v, w1, w2}, and d(v1) = d(w1) = 2,v1v2, w1w2 2 E(G). (iv) There exist two non-neighborly 2-vertices u and v adjacent to a 4-vertex w. Let N(w) = {u, v,w1, w2}, uw1, vw2, w1w2 2 E(G). 3. Main results and proofs In the following proofs, we always assume that the graphs are 2-connected. Theorem 1. Let G be a graph of D(G) = D P 4, q(G) = 2D 1. Then G admits a (D + 2,2)-incidence coloring. Proof. Let v be a maximum degree-vertex of G. Since G is 2-connected, G* = G {v} is also connected. According to the known condition that q(G) = 2D 1, we obtain that D 1 = q(G) D = q(G*) P p(G*) 1 = p(G) 2, and so D P p(G) 1. Based on the fact that D 6 p(G) 1, we obtain D = p(G) 1. Therefore, q(G*) = p(G*) 1. By the definition of tree, G* is a tree. Again according to the construction of G*, G is a 1-tree graph. The conclusion holds by Lemma 4 if G 6¼ S p and by Lemma 3 if G ¼ S p . h Theorem 2. Let G be an outerplanar graph of D(G) = 4. Then G admits a (6, 2)-incidence coloring.
S. Wang et al. / Progress in Natural Science 18 (2008) 575–578
Proof. We proceed the induction on the order p(G) of G. Let C = {1, 2, 3, 4, 5, 6}. When p(G) = 5, G admits a (6, 2)-incidence coloring by the method of enumerating. We suppose that each outerplanar graph of p(G) < p(p P 6) admits a (6, 2)-incidence coloring. Now we consider the incidence coloring of outerplanar graphs of p(G) = p. By Lemma 9, we may divide the proof into four cases as follows. Case 1. There exist two neighborly 2-vertices u and v. Let N(u) = {v, u1}, N(v) = {u, v1}. Let G* = G {u} + u1v. Obviously G* is a 2-connected outerplanar graph of D(G*) = 4 and p(G*) < p. According to the induction hypothesis, G* admits a (6, 2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (6, 2)-incidence coloring r:I(G) ? C of G. Let r(uu1) = r*(vu1). Since jFr(v, vu)j = jIv(G*) [ Av(G*)j 6 4, there exists at least one color available for the coloring of incidence (v, vu). Again since jFr(u, uv)j = j{r(uu1), r(v, vu), r*(v, vv1)}j 6 4, there exists at least one color available for the coloring of incidence (u, uv). The coloring of other incidences in G is the same as r*. Case 2. There exists a 2-vertex u adjacent to a 3-vertex v. Let N(u) = {v, u1}, u1v 2 E(G),N(v) = {u, u1, v1}. Let G* = G {u}. Obviously G* is a 2-connected outerplanar graph of D(G*) 6 4 and p(G*) < p. When D(G*) = 3, by Lemma 6, G* admits a (6, 2)-incidence coloring r*:I(G*) ? C. When D(G*) = 4, according to the induction hypothesis, G* admits a (6, 2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (6, 2)-incidence coloring r:I(G) ? C of G. Let r(u, uu1) = r*(v, vu1), r(u, uv) = r*(u1, u1v). Since jFr(v, vu)j = jIv(G*) [ Av(G*)j 6 4, there exists at least one color available for the coloring of incidence (v,vu). Again since jF r ðu1 ; u1 uÞj ¼ jI u1 ðG Þ[ Au1 ðG Þj 6 5, there exists at least one color available for the coloring of incidence (u1, u1u). The coloring of other incidences in G is the same as r*. Case 3. There exists a 2-vertex u adjacent to two neighborly 4-vertices v and w. Let N(v) = {u, w, v1, v2}, N(w) = {u, v, w1, w2}, d(v1) = d(w1) = 2, v1v2, w1w2 2 E(G). Let G* = G {u}. Obviously G* is a 2-connected outerplanar graph of D(G*) 6 4 and p(G*) < p. When D(G*) = 3, by Lemma 6, G* admits a (6, 2)-incidence coloring r*:I(G*) ? C. When D(G*) = 4, according to the induction hypothesis, G* admits a (6, 2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (6, 2)-incidence coloring r:I(G) ? C of G. Let r(u, uv) = r*(w, wv), and r(u, uw) = r*(v, vw). Since jFr(v, vu)j = jIv(G*) [ Av(G*)j 6 5, there exists at least one color available for the coloring of incidence (v, vu). Again since jFr(w, wu)j = jIw(G*) [ Aw(G*)j 6 5, there exists at least one color available for the coloring of incidence (w, wu). The coloring of other incidences in G is the same as r*. Case 4. There exist two non-neighborly 2-vertices u and v adjacent to a 4-vertex w. Let N(w) = {u, v, w1, w2}, uw1, vw2, w1w2 2 E(G). Let G* = G {u}. Since G is 2-connected, we only need to consider the following two cases.
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Case 4.1. Both w1 and w2 are 3-vertices. We may give a (6, 2)-incidence coloring by the method of enumerating. Case 4.2 Both w1 and w2 are 4-vertices. Obviously G* is a 2-connected outerplanar graph of D(G*) = 4 and p(G*) < p. By the induction hypothesis, G* admits a (6, 2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (6, 2)-incidence coloring r:I(G) ? C of G. Let r(u, uw1) = r*(w, ww1), r(u, uw) = r*(w1, w1w). Since jF r ðw1 ; w1 uÞj ¼ jI w1 ðG Þ[ Aw1 ðG Þj 6 5, there exists at least one color available for the coloring of incidence (w1, w1u). Again since jFr(w, wu)j = jIw(G*) [ Aw(G*)j 6 5, there exists at least one color available for the coloring of incidence (w, wu). The coloring of other incidences in G is the same as r*. From the above mentioned, we obtain that the outerplanar graph G of D(G) = 4 admits a (6, 2)-incidence coloring. h Theorem 3. Let G be an outerplanar graph of D(G) = D. Then G admits a (D + 2,2)-incidence coloring. Proof. We proceed the double inductions on the maximum degree D(G) and the size q(G) of G. Let C = {1, 2, , D + 2}. By Lemma 6 and Theorem 2, the conclusion holds for the condition of D(G) = 3 and 4. Again according to Theorem 1, the conclusion holds for the condition of q(G) = 2D 1. We suppose that each outerplanar graph of D(G) < D(D P 5) or q(G) < q(q P 2D) admits a (D + 2,2)-incidence coloring. By Lemma 9, we may divide the proof into four cases for the outerplanar graph of D(G) = D and q(G) = q. Case 1. There exist two neighborly 2-vertices u and v. Let N(u) = {v, u1}, N(v) = {u, v1}. Let G* = G {u} + u1v. Obviously G* is a 2-connected outerplanar graph of D(G*) = D and q(G*) < q. According to the induction hypothesis, G* admits a (D + 2,2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (D + 2,2)-incidence coloring r:I(G) ? C of G. Let r(uu1) = r*(vu1). Since jFr(v, vu)j = jIv(G*) [ Av(G*)j 6 4, there exists at least one color available for the coloring of incidence (v, vu). Again since jFr(u, uv)j = j{r(uu1), r(v, vu), r*(v, vv1)}j 6 4, there exists at least one color available for the coloring of incidence (u, uv). The coloring of other incidences in G is the same as r*. Case 2. There exists a 2-vertex u adjacent to a 3-vertex v. Let N(u) = {v, u1}, u1v 2 E(G), N(v) = {u, u1, v1}. Let G* = G {u}. Obviously G* is a 2-connected outerplanar graph of D(G*) 6 D and q(G*) < q. According to the induction hypothesis, G* admits a (D + 2,2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (D + 2,2)-incidence coloring r:I(G) ? C of G. Let r(u, uu1) = r*(v, vu1), r(u, uv) = r*(u1, u1v). Since jFr(v, vu)j = jIv(G*) [ Av(G*)j 6 4, there exists at least one color available for the coloring of incidence (v, vu). Again since jF r ðu1 ; u1 uÞj ¼ jI u1 ðG Þ[ Au1 ðG Þj 6 D þ 1, there exists at least one color available for the coloring of incidence (u1, u1u). The coloring of other incidences in G is the same as r*. Case 3. There exists a 2-vertex u adjacent to two neighborly 4-vertices v and w. Let N(v) = {u, w, v1, v2}, N(w) = {u, v, w1, w2}, d(v1) = d(w1) = 2, v1v2, w1w2 2 E(G).
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Let G* = G {u}. Obviously G* is a 2-connected outerplanar graph of D(G*) = D and q(G*) < q. According to the induction hypothesis, G* admits a (D + 2,2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (D + 2,2)-incidence coloring r:I(G) ? C of G. Let r(u, uv) = r*(w, wv), and r(u, uw) = r*(v, vw). Since jFr(v, vu)j = jIv(G*) [ Av(G*)j 6 5, there exists at least one color available for the coloring of incidence (v, vu). Again since jFr(w, wu)j = jIw(G*) [ Aw(G*)j 6 5, there exists at least one color available for the coloring of incidence (w, wu). The coloring of other incidences in G is the same as r*. Case 4. There exist two non-neighborly 2-vertices u and v adjacent to a 4-vertex w. Let N(w) = {u, v, w1, w2}, uw1, vw2, w1w2 2 E(G). Let G* = G {u}. Obviously G* is a 2-connected outerplanar graph of D(G*) 6 D and q(G*) < q. According to the induction hypothesis, G* admits a (D + 2,2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (D + 2,2)-incidence coloring r:I(G) ? C of G. Let r(u, uw1) = r*(w, ww1), and r(u, uw) = r*(w1, w1w). Since jF r ðw1 ; w1 uÞj ¼ jI w1 ðG Þ [ Aw1 ðG Þj 6 D þ 1, there exists at least one color available for the coloring of incidence (w1, w1u). Again since jFr(w, wu)j = jIw(G*) [ Aw(G*)j 6 5, there exists at least one color available for the coloring of incidence (w, wu). The coloring of other incidences in G is the same as r*. From the above mentioned, we obtain that there exists a (D + 2,2)-incidence coloring in the outerplanar graph G of D(G) = D. h Acknowledgments This work was supported by National Natural Science Foundation of China (Grant Nos. 60503002, 60403002,
60403001) and Postdoctoral Science Foundation of China (Grant No. 20060400344). References [1] Brualdi RA, Quinn Massey JJ. Incidence and strong edge colorings graphs. Discrete Math 1993;122:51–8. [2] Guiduli B. On incidence coloring and star arboricity of graphs. Discrete Math 1997;163:275–8. [3] Algor I, Alon N. The star arboricity of graphs. Discrete Math 1989;75:11–22. [4] Chen DL, Liu XK, Wang SD. The incidence coloring number of graph and the incidence coloring conjecture. Math Econ 1998;15(3):47–51, [in Chinese]. [5] Shiu WC, Lam PCB, Chen DL. On incidence coloring for some cubic graphs. Discrete Math 2002;252:259–66. [6] Duan H, Chen DL. The incidence chromatic number of 1-tree graphs. J Shandong Univ Sci Technol 2002;21(1):33–41, [in Chinese]. ´ ric Sopena E, Zhu XD. Incidence coloring of k[7] Dolama MH, E degenerated graphs. Discrete Math 2004;283:121–8. [8] Maydanskiy M. The incidence coloring conjecture for graphs of maximum degree 3. Discrete Math 2005;292:131–41. [9] Shiu WC, Sun PK. Graphs which are not (D + 1)-incidence colorable with erratum to the incidence chromatic number of outerplanar graphs. Department of Mathematics Hong Kong Baptist University, Technical Report; 2006;419:1–8. [10] Wang SD, Chen DL, Pang SC. The incidence coloring of Halin graphs and outerplanar graphs. Discrete Math 2002;256:397–405. [11] Qu JZ. Studies on the coloring of some graphs. Master thesis. Shandong Inst Mining Technol; 1998 [in Chinese]. [12] Bondy JA, Murty USR. Graph theory with applications. New York: Macmillan; 1976. [13] Chen DL, Wu JL, Wang SD. A structural theorem of outerplanar graphs. J Shandong Inst Mining Technol 1999;18(4):44–6, [in Chinese].
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. 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 The concept of incidence coloring was introduced by Brualdi and Massey in 1993 [1]. Let G = (V, E) be a multigraph of order p and size q, I(G) = {(v, e)jv 2 V,e 2 E and v is incident with e} be the set of incidences of G. We say that two incidences (v, e) and (w, f) are neighborly if one of the following conditions holds: (i) v = w; (ii) e = f; (iii) the edge vw equals e or f. An incidence coloring of graph G is a coloring of its incidences in which neighborly incidences are assigned different colors. The incidence chromatic number of G, denoted by vi(G), is the smallest number k of colors such that there exists a k-incidence coloring in G. Brualdi and Massey [1] determined the incidence chromatic numbers of trees,
*
Corresponding author. Tel.: +86 13681078897. E-mail address: [email protected] (S. Wang).
complete graphs and complete bipartite graphs, and put forward the incidence coloring conjecture (ICC). They supposed that the incidence chromatic number vi(G) is always less than or equal to the maximum degree of the graph plus 2. Guiduli [2] found that the incidence coloring is a special case of directed star arboricity introduced by Algor and Alon [3], and he also proved that ICC is incorrect. Moreover, according to a tight upper bound for directed star arboricity, Guiduli gave an upper bound for incidence chromatic number, vi(G) 6 D + O(log D). Although ICC has been proved incorrect, how to determine the incidence chromatic number of graph is fascinating. Chen et al. [4] determined the incidence chromatic numbers of paths, cycles, fans, wheels, adding-edge wheels and complete 3partite graphs. Then, the incidence chromatic numbers of many types of graphs were determined [5–8]. Later, Shiu and Sun [9] gave a counter example to show that outerplanar graph of D = 4 is not 5-incidence colorable. This contradicts the incidence chromatic number of outerplanar graphs proved in Ref. [10]. This contradiction results from using the following Statement 1 several times which, however, is incorrect in the proof of the incidence chromatic number of outerplanar graphs [10]. Thus, the proof of
1002-0071/$ - see front matter 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 should be void and the problem of determining the incidence chromatic number of 2-connected outerplanar graphs is still open. 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. Statement 1. [11] Let v be a cut-vertex of outerplanar graph G, G v = H1 [ H2, G1 = G[V(H1) [ {v}], G2 = G[V(H2) [ {v}]. Then
Lemma 4. [6] Graph S p admits a (p, 1)-incidence coloring.
vi ðGÞ ¼ maxfvi ðG1 Þ; vi ðG2 Þ; d G ðvÞ þ 1g:
Lemma 8. Let G be a 2-connected graph. Then q(G) P 2D(G) 1.
2. Lemmas and notations In this study, we always limit to finite, simple, and undirected graphs. In a given graph G, the vertex of degree k is 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 adjacent to v of G, respectively, and p, q, D, N(v) for short, respectively. For simplicity, we denote the set of all the incidences of the form (u, uv) and (v, vu) of G by Av(G) and Iv(G) (abbreviated as Av and Iv), respectively, where u is an adjacent-vertex of v. Let r be a k-incidence coloring 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 color set for the incidence (u, uv) in r, namely, Fr(u, uv) = r(Iu) [ r(Au) [ r(Iv). In a k-incidence coloring r of G, if 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 can be found in Ref. [12]. In addition, we need the following lemmas to obtain the main results. Lemma 1. Each 2-connected graph is 2-edge-connected. Proof. (disproof) Suppose that G is 2-connected but not 2edge-connected. Then there exists an edge e = uv in G such that G e is disconnected. Thus G u is also disconnected. That is to say, u is a cut-vertex of G. This contradicts that G is 2-connected graph. The proof is completed. h Let G = (V, E) be a planar graph. If $v 2 V(G) such that G v is a forest, then we call G 1-tree graph. We denote by S p the graph with p vertices u, v, x1, x2, , xp2 and 2p 4 edges ux1, ux2, uxp2, vx1, vx2, , vxp2. Lemma 2. [6] Let G be a 2-edge-connected 1-tree graph of D(G) = D P 4 and G 6¼ S p . Then vi(G) = D + 1. According to Lemmas 1 and 2, we may obtain the following Lemma 3. Lemma 3. Let G be a 2-connected 1-tree graph of D(G) = D P 4 and G 6¼ S p . Then vi(G) = D + 1.
Lemma 5. [8] Let G be a graph of D(G) = 3. Then vi(G) 6 5. Lemma 6. Let G be an outerplanar graph of D(G) = 3. Then there exists a (5, 2)-incidence coloring in G. Proof. According to Lemma 5, for "v 2 V(G), there are at most two colors in Av. Hence, the above lemma holds. h Lemma 7. [12] Let G be a connected graph. Then q(G) P p(G) 1.
Proof. Let v be a maximum degree-vertex of G. Let G* = G v. By the 2-connectivity of G, G* is also connected. According to Lemma 7, q(G*) P p(G*) 1. Because q(G*) = q(G) D(G), p(G*) = p(G) 1, q(G) = q(G*) + D(G) P p(G*) 1 + D(G) = p(G) 2 + D(G). Again since G is a simple graph, p(G) 1 P D(G). Therefore, q(G) P 2D(G) 1. h Lemma 9. [13] Let G be a 2-connected outerplanar graph of order p(G) P 3. Then one of the following conditions holds: (i) There exist two neighborly 2-vertices u and v. (ii) There exists a 2-vertex u adjacent to a 3-vertex v. Let N(u) = {v, u1}, vu1 2 E(G). (iii) There exists a 2-vertex u adjacent to two neighborly 4vertices v and w. Let N(v) = {u, w, v1, v2},N(w) = {u, v, w1, w2}, and d(v1) = d(w1) = 2,v1v2, w1w2 2 E(G). (iv) There exist two non-neighborly 2-vertices u and v adjacent to a 4-vertex w. Let N(w) = {u, v,w1, w2}, uw1, vw2, w1w2 2 E(G). 3. Main results and proofs In the following proofs, we always assume that the graphs are 2-connected. Theorem 1. Let G be a graph of D(G) = D P 4, q(G) = 2D 1. Then G admits a (D + 2,2)-incidence coloring. Proof. Let v be a maximum degree-vertex of G. Since G is 2-connected, G* = G {v} is also connected. According to the known condition that q(G) = 2D 1, we obtain that D 1 = q(G) D = q(G*) P p(G*) 1 = p(G) 2, and so D P p(G) 1. Based on the fact that D 6 p(G) 1, we obtain D = p(G) 1. Therefore, q(G*) = p(G*) 1. By the definition of tree, G* is a tree. Again according to the construction of G*, G is a 1-tree graph. The conclusion holds by Lemma 4 if G 6¼ S p and by Lemma 3 if G ¼ S p . h Theorem 2. Let G be an outerplanar graph of D(G) = 4. Then G admits a (6, 2)-incidence coloring.
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Proof. We proceed the induction on the order p(G) of G. Let C = {1, 2, 3, 4, 5, 6}. When p(G) = 5, G admits a (6, 2)-incidence coloring by the method of enumerating. We suppose that each outerplanar graph of p(G) < p(p P 6) admits a (6, 2)-incidence coloring. Now we consider the incidence coloring of outerplanar graphs of p(G) = p. By Lemma 9, we may divide the proof into four cases as follows. Case 1. There exist two neighborly 2-vertices u and v. Let N(u) = {v, u1}, N(v) = {u, v1}. Let G* = G {u} + u1v. Obviously G* is a 2-connected outerplanar graph of D(G*) = 4 and p(G*) < p. According to the induction hypothesis, G* admits a (6, 2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (6, 2)-incidence coloring r:I(G) ? C of G. Let r(uu1) = r*(vu1). Since jFr(v, vu)j = jIv(G*) [ Av(G*)j 6 4, there exists at least one color available for the coloring of incidence (v, vu). Again since jFr(u, uv)j = j{r(uu1), r(v, vu), r*(v, vv1)}j 6 4, there exists at least one color available for the coloring of incidence (u, uv). The coloring of other incidences in G is the same as r*. Case 2. There exists a 2-vertex u adjacent to a 3-vertex v. Let N(u) = {v, u1}, u1v 2 E(G),N(v) = {u, u1, v1}. Let G* = G {u}. Obviously G* is a 2-connected outerplanar graph of D(G*) 6 4 and p(G*) < p. When D(G*) = 3, by Lemma 6, G* admits a (6, 2)-incidence coloring r*:I(G*) ? C. When D(G*) = 4, according to the induction hypothesis, G* admits a (6, 2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (6, 2)-incidence coloring r:I(G) ? C of G. Let r(u, uu1) = r*(v, vu1), r(u, uv) = r*(u1, u1v). Since jFr(v, vu)j = jIv(G*) [ Av(G*)j 6 4, there exists at least one color available for the coloring of incidence (v,vu). Again since jF r ðu1 ; u1 uÞj ¼ jI u1 ðG Þ[ Au1 ðG Þj 6 5, there exists at least one color available for the coloring of incidence (u1, u1u). The coloring of other incidences in G is the same as r*. Case 3. There exists a 2-vertex u adjacent to two neighborly 4-vertices v and w. Let N(v) = {u, w, v1, v2}, N(w) = {u, v, w1, w2}, d(v1) = d(w1) = 2, v1v2, w1w2 2 E(G). Let G* = G {u}. Obviously G* is a 2-connected outerplanar graph of D(G*) 6 4 and p(G*) < p. When D(G*) = 3, by Lemma 6, G* admits a (6, 2)-incidence coloring r*:I(G*) ? C. When D(G*) = 4, according to the induction hypothesis, G* admits a (6, 2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (6, 2)-incidence coloring r:I(G) ? C of G. Let r(u, uv) = r*(w, wv), and r(u, uw) = r*(v, vw). Since jFr(v, vu)j = jIv(G*) [ Av(G*)j 6 5, there exists at least one color available for the coloring of incidence (v, vu). Again since jFr(w, wu)j = jIw(G*) [ Aw(G*)j 6 5, there exists at least one color available for the coloring of incidence (w, wu). The coloring of other incidences in G is the same as r*. Case 4. There exist two non-neighborly 2-vertices u and v adjacent to a 4-vertex w. Let N(w) = {u, v, w1, w2}, uw1, vw2, w1w2 2 E(G). Let G* = G {u}. Since G is 2-connected, we only need to consider the following two cases.
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Case 4.1. Both w1 and w2 are 3-vertices. We may give a (6, 2)-incidence coloring by the method of enumerating. Case 4.2 Both w1 and w2 are 4-vertices. Obviously G* is a 2-connected outerplanar graph of D(G*) = 4 and p(G*) < p. By the induction hypothesis, G* admits a (6, 2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (6, 2)-incidence coloring r:I(G) ? C of G. Let r(u, uw1) = r*(w, ww1), r(u, uw) = r*(w1, w1w). Since jF r ðw1 ; w1 uÞj ¼ jI w1 ðG Þ[ Aw1 ðG Þj 6 5, there exists at least one color available for the coloring of incidence (w1, w1u). Again since jFr(w, wu)j = jIw(G*) [ Aw(G*)j 6 5, there exists at least one color available for the coloring of incidence (w, wu). The coloring of other incidences in G is the same as r*. From the above mentioned, we obtain that the outerplanar graph G of D(G) = 4 admits a (6, 2)-incidence coloring. h Theorem 3. Let G be an outerplanar graph of D(G) = D. Then G admits a (D + 2,2)-incidence coloring. Proof. We proceed the double inductions on the maximum degree D(G) and the size q(G) of G. Let C = {1, 2, , D + 2}. By Lemma 6 and Theorem 2, the conclusion holds for the condition of D(G) = 3 and 4. Again according to Theorem 1, the conclusion holds for the condition of q(G) = 2D 1. We suppose that each outerplanar graph of D(G) < D(D P 5) or q(G) < q(q P 2D) admits a (D + 2,2)-incidence coloring. By Lemma 9, we may divide the proof into four cases for the outerplanar graph of D(G) = D and q(G) = q. Case 1. There exist two neighborly 2-vertices u and v. Let N(u) = {v, u1}, N(v) = {u, v1}. Let G* = G {u} + u1v. Obviously G* is a 2-connected outerplanar graph of D(G*) = D and q(G*) < q. According to the induction hypothesis, G* admits a (D + 2,2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (D + 2,2)-incidence coloring r:I(G) ? C of G. Let r(uu1) = r*(vu1). Since jFr(v, vu)j = jIv(G*) [ Av(G*)j 6 4, there exists at least one color available for the coloring of incidence (v, vu). Again since jFr(u, uv)j = j{r(uu1), r(v, vu), r*(v, vv1)}j 6 4, there exists at least one color available for the coloring of incidence (u, uv). The coloring of other incidences in G is the same as r*. Case 2. There exists a 2-vertex u adjacent to a 3-vertex v. Let N(u) = {v, u1}, u1v 2 E(G), N(v) = {u, u1, v1}. Let G* = G {u}. Obviously G* is a 2-connected outerplanar graph of D(G*) 6 D and q(G*) < q. According to the induction hypothesis, G* admits a (D + 2,2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (D + 2,2)-incidence coloring r:I(G) ? C of G. Let r(u, uu1) = r*(v, vu1), r(u, uv) = r*(u1, u1v). Since jFr(v, vu)j = jIv(G*) [ Av(G*)j 6 4, there exists at least one color available for the coloring of incidence (v, vu). Again since jF r ðu1 ; u1 uÞj ¼ jI u1 ðG Þ[ Au1 ðG Þj 6 D þ 1, there exists at least one color available for the coloring of incidence (u1, u1u). The coloring of other incidences in G is the same as r*. Case 3. There exists a 2-vertex u adjacent to two neighborly 4-vertices v and w. Let N(v) = {u, w, v1, v2}, N(w) = {u, v, w1, w2}, d(v1) = d(w1) = 2, v1v2, w1w2 2 E(G).
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Let G* = G {u}. Obviously G* is a 2-connected outerplanar graph of D(G*) = D and q(G*) < q. According to the induction hypothesis, G* admits a (D + 2,2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (D + 2,2)-incidence coloring r:I(G) ? C of G. Let r(u, uv) = r*(w, wv), and r(u, uw) = r*(v, vw). Since jFr(v, vu)j = jIv(G*) [ Av(G*)j 6 5, there exists at least one color available for the coloring of incidence (v, vu). Again since jFr(w, wu)j = jIw(G*) [ Aw(G*)j 6 5, there exists at least one color available for the coloring of incidence (w, wu). The coloring of other incidences in G is the same as r*. Case 4. There exist two non-neighborly 2-vertices u and v adjacent to a 4-vertex w. Let N(w) = {u, v, w1, w2}, uw1, vw2, w1w2 2 E(G). Let G* = G {u}. Obviously G* is a 2-connected outerplanar graph of D(G*) 6 D and q(G*) < q. According to the induction hypothesis, G* admits a (D + 2,2)-incidence coloring r*:I(G*) ? C. Now we extend r* to a (D + 2,2)-incidence coloring r:I(G) ? C of G. Let r(u, uw1) = r*(w, ww1), and r(u, uw) = r*(w1, w1w). Since jF r ðw1 ; w1 uÞj ¼ jI w1 ðG Þ [ Aw1 ðG Þj 6 D þ 1, there exists at least one color available for the coloring of incidence (w1, w1u). Again since jFr(w, wu)j = jIw(G*) [ Aw(G*)j 6 5, there exists at least one color available for the coloring of incidence (w, wu). The coloring of other incidences in G is the same as r*. From the above mentioned, we obtain that there exists a (D + 2,2)-incidence coloring in the outerplanar graph G of D(G) = D. h Acknowledgments This work was supported by National Natural Science Foundation of China (Grant Nos. 60503002, 60403002,
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