### Linear arboricity of graphs embedded in a surface of non-negative Euler characteristic

CHEN Hong-ling, WANG Hui-juan*, GAO Hong-wei

1. School of Mathematics and Statistics, Qingdao University, Qingdao 266071, Shandong, China
• Online:2018-12-20 Published:2018-12-18

Abstract: The linear arboricity of graph G, denoted by la(G), is the minimum number of linear forest required to partition the edge set E(G), which is an improper edge coloring. The linear arboricity of graph which can be embedded in a surface of non-negative Euler characteristic with maximum Δ(G)≥7 is mainly studied. If there is no adjacent chordal 6-cycle, then the arboricity of graph G is「Δ/2.

Key words: Euler characteristic, linear forest, cycle

CLC Number:

• O157.5
 [1] FRANK H. Covering and packing in graphs, I.[J]. Annals of the New York Academy of Sciences, 2010, 175(1):198-205.[2] AKIYAMA J, EXOO G, HARARY F. Covering and packing in graphs III: cyclic and acyclic invariants[J]. Mathematica Slovaca, 1980, 30(4):405-417.[3] WU J L, LIU G, WU Y W. The linear arboricity of composition graphs[J]. Journal of Systems Science & Complexity, 2002, 15(4):38-41.[4] WU Jianling. The linear arboricity of Series-Parallel graphs[J]. Graphs & Combinatorics, 2000, 16(3):367-372.[5] JIN A, EXOO G, HARARY F. Covering and packing in graphs IV: linear arboricity[J]. Networks, 2010, 11(1):69-72.[6] ENMOTO H, PEROCHE B. The linear arboricity of some regular graphs[J]. Journal of Graph Theory, 1984, 8(2):309-324.[7] GULDAN F. The linear arboricity of 10 regular graph[J]. Mathematical Institute of the Slovak Academy of Sciences, 1986, 36(3):225-228.[8] WU Jianliang. On the linear arboricity of planar graphs[J]. Journal of Graph Theory, 1999, 31(2):129-134.[9] WU Jianliang, WU Yuwen. The linear arboricity of planar graphs of maximum degree seven are four[J]. Journal of Graph Theory, 2008, 58(3):210-220.[10] CYGAN M, HOU J, KOWALIK L, et al. A planar linear arboricity conjecture[J]. Journal of Graph Theory, 2012, 69(4):403-425.[11] WANG Huijuan, WU Lidong, WU Weili, et al. Minimum number of disjoint linear forests covering a planar graph[J]. Journal of Combinatorial Optimization, 2014, 28(1):274-287.[12] ALON N. The linear arboricity of graphs[J]. Israel Journal of Mathematics, 1988, 62(3):311-325.[13] WU Jianliang, HOU Jianfeng, SUN Xiangyong. A note on the linear arboricity of planar graphs without 4-cycles[J]. The Eighth International Symposium on Operations Research and Its Applications, 2009, 09:174-178.
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