List edge and list total colorings of planar graphs without non-induced 7-cycles

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1 List edge and ist tota coorings of panar graphs without non-induced 7-cyces Aijun Dong, Guizhen Liu, Guojun Li To cite this version: Aijun Dong, Guizhen Liu, Guojun Li. List edge and ist tota coorings of panar graphs without non-induced 7-cyces. Discrete Mathematics and Theoretica Computer Science, DMTCS, 013, Vo. 15 no. 1 (1), pp <ha > HAL Id: ha Submitted on 13 May 014 HAL is a muti-discipinary open access archive for the deposit and dissemination of scientific research documents, whether they are pubished or not. The documents may come from teaching and research institutions in France or abroad, or from pubic or private research centers. L archive ouverte puridiscipinaire HAL, est destinée au dépôt et à a diffusion de documents scientifiques de niveau recherche, pubiés ou non, émanant des étabissements d enseignement et de recherche français ou étrangers, des aboratoires pubics ou privés.

2 Discrete Mathematics and Theoretica Computer Science DMTCS vo. 15:1, 013, List Edge and List Tota Coorings of Panar Graphs without non-induced 7-cyces Aijun Dong 1 Guizhen Liu Guojun Li 1 Schoo of science, Shandong Jiaotong University, Jinan 5003, P. R. China Schoo of Mathematics, Shandong University, Jinan 50100, P. R. China received nd November 010, revised 11 th October 011, 10 th Juy 01, accepted 19 th February 013. Giving a panar graph G, et χ (G) and χ (G) denote the ist edge chromatic number and ist tota chromatic number of G respectivey. It is proved that if G is a panar graph without non-induced 7-cyces, then χ (G) (G) + 1 and χ (G) (G) + where (G) 7. Keywords: List cooring; Panar graph; Choosabiity. 1 Introduction The terminoogy and notation used but undefined in this paper can be found in [1]. Let G be a graph and we use V (G), E(G), F (G), (G) and δ(g) to denote the vertex set, edge set, face set, maximum degree, and minimum degree of G, respectivey. Let d G (x) or simpy d(x), denote the degree of a vertex (resp. face) x in G. A vertex (resp. face) x is caed a k-vertex (resp. k-face), k + -vertex (resp. k + -face), or k -vertex, if d(x) = k, d(x) k, or d(x) k. We use (d 1, d,, d n ) to denote a face f if d 1, d,, d n are the degrees of vertices which are incident with the face f. If u 1, u,, u n are the vertices on the boundary wak of a face f, then we write f = u 1 u u n. Let δ(f) denote the minima degree of vertices which are incident with f. We use f i (v) to denote the number of i-faces which are incident with v for each v V (G). Let n i (f) denote the number of i-vertices which are incident with f for each f F (G). A cyce C of ength k is caed k-cyce, and if there is at east one edge xy E(G)\E(C) and x, y V (C), the cyce C is caed non-induced k-cyce. The mapping L is said to be a tota assignment for a graph G if it assigns a ist L(x) of possibe coors to each eement x V (G) E(G). If G has a proper tota cooring φ(x) L(x) for a x V (G) E(G), then we say that G is tota-l-coorabe. Let f : V (G) E(G) N where f is a function into the positive integers. We say that G is tota-f-choosabiity if it is tota-l-coorabe for every tota assignment L Supported by NSFC grants , E-mai: dongaijun@mai.sdu.edu.cn E-mai: gziu@sdu.edu.cn. E-mai: guojunsdu@gmai.com c 013 Discrete Mathematics and Theoretica Computer Science (DMTCS), Nancy, France

3 10 A. J. Dong, G. Z. Liu, G. J. Li satisfying L(x) = f(x) for a x V (G) E(G). The ist tota cooring number χ (G) of G is the smaest integer k such that G is tota-f-choosabiity when f(x) = k for each x V (G) E(G). The ist edge cooring number χ (G) of G is defined simiary in terms of cooring edges aone; and so is the concept of edge-f-choosabiity. On the ist cooring number of a graph G, there is a famous conjecture known as the List Cooring Conjecture. Conjecture 1 For a mutigraph G, (a) χ (G) = χ (G); (b) χ (G) = χ (G). Part (a) of Conjecture 1 was formuated independenty by Vizing, by Gupta, by Aberson and Coins, and by Boobás and Harris [6, 11]. It is we known as the List Cooring Conjecture. Part (b) was formuated by Borodin, Kostochka and Wooda []. Part (a) has been proved for bipartite mutigraphs [5]. Part (a) and Part (b) have been proved for outerpanar graphs [15], and graphs with 1 which can be embedded in a surface of nonnegative characteristic []. There are severa reated resuts for panar graphs, such as panar graphs without 4-cyces by Hou et a.[9], panar graphs without 4- and 5-cyces or panar graphs without intersecting 4-cyces by Liu et a.[13], panar graphs without trianges adjacent 4-cyces by Li et a.[14], panar graphs without intersecting trianges by Sheng et a.[18]. To confirm Conjecture 1 is a chaenging work. From the Vizing Theorem and the Tota Cooring Conjecture, the foowing weak conjecture is presented. Conjecture For a mutigraph G, (a) χ (G) (G) + 1; (b) χ (G) (G) +. Part (a) of Conjecture has been proved for compete graphs of odd order [7]. Wang et a. confirmed part (a) of Conjecture for panar graphs without 6-cyces or without 5-cyces [17, 16]. Zhang et a. proved part (a) of Conjecture for panar graphs without trianges [19]. Hou et a. proved part (a) of Conjecture for panar graphs without adjacent trianges or 7-cyces [8]. Cai et a. confirmed part (a) of Conjecture for panar graphs without chorda 5-cyces [3]. Part (b) of Conjecture was proved by Hou et a. for panar graphs G with (G) 9 [10]. Dong et a. confirmed Conjecture for panar graphs without 6-cyces with chord [4]. In this paper, we sha show the foowing resut. Theorem Let G be a panar graph without non-induced 7-cyces, if (G) 7, then χ (G) (G) + 1 and χ (G) (G) +. Panar graphs without non-induced 7-cyces First et us introduce some important emmas. Lemma 3 Let G be a panar graph without non-induced 7-cyces. Then there is an edge uv E(G) such that min{d(u), d(v)} (G)+1 and d(u) + d(v) max{9, (G) + }. Proof: Suppose to the contrary that G is a minima counterexampe to Lemma 3 in terms of the number of vertices and edges. Then we have δ(g) 3.

4 List Edge and List Tota Coorings 103 By Euer s formua V E + F = and v V (G) d(v) = f F (G) d(f) = E, we have (d(v) 6) + (d(f) 6) = 6( V E + F ) = 1. v V (G) f F (G) Define an initia charge function w on V (G) F (G) by setting w(v) = d(v) 6 if v V (G) and w(f) = d(f) 6 if f F (G), so that x V (G) F (G) w(x) = 1. Now redistribute the charge according to the foowing discharging rues. For convenience, et w(v) denote the tota charge transferred from a vertex v to a its incident 4- and 5-faces where d(v) = 5. D1 Let f be a 3-face incident with a vertex v. Then v gives f charge if d(v) = 5, 3 if d(v) 6. D Let f be a 4-face incident with a vertex v. Then v gives f charge 1 if d(v) = 4, 5 and 6, 1 if d(v) 7. D3 Let f be a 5-face incident with a vertex v. Then v gives f charge 1 5 if d(v) = 4, 5 and 6, 1 3 if d(v) 7. Let the new charge of each eement x be w (x) for each x V (G) F (G). In the foowing, et us check the new charge w (x) of each eement x V (G) F (G). Suppose d(v) = 3. Then w (v) = w(v) = 0. Suppose d(v) = 4. Then w(v) =, f 4 (v) 4. If f 4 (v) 4, then f 5 (v) = 0 for G contains no non-induced 7-cyces. We have w (v) 1 4 = 0 by D. Otherwise, i.e. f 4(v) 1, then f 5 (v) 4. Thus we have w (v) > = 7 10 > 0 by D and D3. Suppose d(v) = 5. Then w(v) = 4, f 3 (v) 5. If 1 f 3 (v), then w (v) 4 w(v) = 0 by D1. Otherwise, i.e. f 3 (v) = 0, then f 4 (v) + f 5 (v) 5. It is cear that w (v) > = 3 > 0 by D and D3. Suppose d(v) = 6. Then w(v) = 6, f 3 (v) 4 for G contains no non-induced 7-cyces. If f 3 (v) = 4, then f 4 (v) = 0 and f 5 (v) = 0 for G contains no non-induced 7-cyces. We have w (v) = 0 by D1. If f 3 (v) 3, then it is cear that w (v) > = 0 by D1, D and D3. Suppose d(v) = 7. Then w(v) = 8, f 3 (v) 5 for G contains no non-induced 7-cyces. Suppose f 3 (v) = 5. Then f 4 (v) = 0 and f 5 (v) = 0 for G contains no non-induced 7-cyces. We can get w (v) = 1 > 0 by D1. Suppose f 3 (v) = 4. Then f 4 (v). If f 4 (v) =, then f 5 (v) = 0 for G contains no non-induced 7-cyces. We have w (v) = 0 by D1 and D. If f 4(v) 1, then f 5 (v) 1 for G contains no non-induced 7-cyces. We have w (v) = 3 > 0 by D1, D and D3. Suppose f 3 (v) = 3. Then f 4 (v) and f 5 (v) for G contains no non-induced 7-cyces. It is cear that w (v) > = 5 6 > 0 by D1, D and D3. Suppose f 3 (v). Then it is cear that w (v) > = 0 by D1, D and D3. Suppose d(v) = 8. Then w(v) = 10, f 3 (v) 6 for G contains no non-induced 7-cyces. If f 3 (v) = 6, then f 4 (v) = 0 and f 5 (v) = 0 for G contains no non-induced 7-cyces. We have w (v) = 1 > 0 by D1. If f 3 (v) = 5, then f 4 (v) 1 and f 5 (v) 1 for G contains no non-induced 7-cyces. We can get w (v) = 7 6 > 0 by D1, D and D3. If 4, then it is cear that w (v) = 0 by D1, D and D3.

5 104 A. J. Dong, G. Z. Liu, G. J. Li Suppose d(v) = 9. Then w(v) = 1, f 3 (v) 7 for G contains no non-induced 7-cyces. If f 3 (v) = 7, then f 4 (v) = 0 and f 5 (v) = 0 for G contains no non-induced 7-cyces. We can get w (v) = 3 > 0 by D1. If 6, then it is cear that w (v) > = 0 by D1, D and D3. Suppose d(v) 10. Then w(v) = d(v) 6, f 4 (v) + f 5 (v) d(v) f 3 (v). Thus we have w (v) d(v) 6 3 f 4 (v) 1 3 f 5(v) d(v) 6 1 by D1, D and D3. Since f 3 (v) 4 5 d(v), we have w (v) 3 5d(v) 6 0. Suppose d(f) = 3. Then w(f) = 3. Suppose δ(f) = 3. Then f is a (3, 7 +, 7 + )-face by assumption. We have w (f) = = 0 by D1. Suppose δ(f) = 4. Then f is a (4, 6 +, 6 + )-face by assumption. We have w (f) = = 0 by D1. Suppose δ(f) = 5. Then f is a (5, 5 +, 5 + )-face. Suppose f is a (5, 5, 5)-face. For convenience, et f = uvw. Of the three vertices u, v and w, there is at most one vertex which is incident with at east four 3-faces for the reason that G contains no non-induced 7-cyces. Without oss of generaity, et f 3 (u) 4. Then f 3 (u) + f 4 (u) + f 5 (u) 5, f 3 (v) + f 4 (v) + f 5 (v) 3 and f 3 (w)+f 4 (w)+f 5 (w) 3 for G contains no non-induced 7-cyces. We have f 3(u) 4 5, 4 3, f 3(w) 4 3 by D and D3. Thus w (f) = 7 15 > 0 by D1. Now we assume that f 3 (u) 3, f 3 (v) 3, f 3 (w) 3. Then we have 1 for G contains no non-induced 7-cyces and by D1, D and D3. Thus w (f) = 0 by D1. Suppose f is a (5, 5, 6 + )-face. For convenience, et f = uvw where d(u) = d(v) = 5. Since f 3 (u) + f 4 (u) + f 5 (u) 5, f 3 (v) + f 4 (v) + f 5 (v) 5, we have by D and f 3(u) 4 5, D3. Thus w (f) = 1 10 > 0 by D1. Suppose f is a (5, 6 +, 6 + )-face. Then we have w (f) > = 0 by D1. Suppose δ(f) 6. Then we have w (f) = = 3 > 0 by D1. Suppose d(f) = 4. Then w(f) =. If δ(f) = 3, then f is a (3, 3 +, 7 +, 7 + )-face by assumption. We have w (f) + 1 = 0 by D. If δ(f) 4, then f is a (4 +, 4 +, 4 +, 4 + )-face. We have w (f) = 0 by D. Suppose d(f) = 5. Then w(f) = 1. Suppose δ(f) = 3. Then n 3 (f) by assumption. If n 3 (f) =, then f is a (3, 3, 7 +, 7 +, 7 + )-face by assumption. We have w (f) = 0 by D3. If n 3(f) = 1, then f is a (3, 4 +, 4 +, 7 +, 7 + )-face by assumption. We have w (f) = 1 15 > 0 by D3. Suppose δ(f) 4. Then we have w (f) = 0 by D3. Suppose d(f) 6. Then w (f) = w(f) 0. From the above discussion, we obtain 1 = x V (G) F (G) w (x) 0, a contradiction. Lemma 4 Let G be a panar graph without non-induced 7-cyces. Then χ (G) k + 1 and χ (G) k + where k = max{ (G), 7}. Proof: Suppose to the contrary that G and G are minima counterexampes to the concusions for χ and χ respectivey. Let L and L be ist assignments such that L (e) = k + 1 for each e E(G), G is not edge-l -coorabe, and L (x) = k + for each x V (G) E(G), G is not tota-l -coorabe. By Lemma 3, G and G contain an edge uv E(G) such that min{d(u), d(v)} (G)+1 and d(u) + d(v) max{9, (G) + } = k

6 List Edge and List Tota Coorings 105 Let Ḡ = G uv. Then Ḡ is edge-l -coorabe by assumption. For d(u) + d(v) k +, there are at most k edges which are adjacent to uv in Ḡ. Thus there is at est one coor in L (uv) which we can use to coor uv. Then G is edge-l -coorabe, a contradiction. Let G = G uv. Then G is tota-l -coorabe by assumption. With oss of generaity, et d(u) = min{d(u), d(v)}. Erase the coor on u, then there is at east one coor in L (uv) which we can use to coor uv for d(u) + d(v) k +. For d(u) (G)+1 k+1 k+1, then u is adjacent to at most vertices and is incident with at most k+1 edges. Thus there is at east one coor in L (u) which we can use to coor u. Then G is tota-l -coorabe, a contradiction. From the above discussion, we have χ (G) k + 1 and χ (G) k + where k = max{ (G), 7}. By Lemma 4, it is easy to obtain the main theorem. Theorem Let G be a panar graph without non-induced 7-cyces, if (G) 7, then χ (G) (G) + 1 and χ (G) (G) +. References [1] J. A. Bondy and U. S. R. Murty, Graph Theory with Appications, North-Hoand, New York, [] O. Borodin, A. Kostochka, D. Wooda, List edge and ist tota coourings of mutigraphs, J. Combin. Theory Ser. B 71 (1997) [3] J. Cai, J. Hou, X. Zhang, G. Liu, Edge-choosabiity of panar graphs without non-induced 5-cyces, Information Processing Letters, 109 (7) (009) [4] A. DONG, G. Liu, G. Li, List edge and ist tota coorings of panar graphs without 6-cyces with chord, Bu. Korean Math. Soc. 49 (01) [5] F. Gavin, The ist chromatic index of a bipartite mutigraph, J. Combin. Theory Ser. B 63 (1995) [6] R. Hägkvist, A. Chetwynd, Some upper bounds on the tota and ist chromatic numbers of mutigraphs, J. Graph Theory 16 (199) [7] R. Hägkvist, J, Janssen, New bounds on the ist-chromatic index of the compete graph and other simpe graphs, Combin. Probab. Comput. 6 (1997) [8] J. Hou, G. Liu, J. Cai, Edge-choosabiity of panar graphs without adjacent trianges or without 7-cyce, Discrete Mathematics, 309 (009) [9] J. Hou, G. Liu, J. Cai, List Edge and List Tota Coorings of Panar Graphs without 4-cyces, Theoretica Computer Science, 369 (006) [10] J. Hou, G. Liu, J. Wu, Some resuts on ist tota coorings of panar graphs, Lecture Note in Computer Science, 4489 (007) [11] T. Jensen, B. Toft, Graph Cooring Probem, Wiey-Interscience, New York, 1995.

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