JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (11): 147-154.doi: 10.6040/j.issn.1671-9352.0.2022.221

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Linear arboricity on embedded graphs without adjacent short cycles

Hongjun DU(),Huijuan WANG*()   

  1. School of Mathematics and Statistics, Qingdao University, Qingdao 266071, Shandong, China
  • Received:2022-04-18 Online:2023-11-20 Published:2023-11-07
  • Contact: Huijuan WANG E-mail:dhjqdu@163.com;sduwhj@163.com

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Abstract:

Linear arboricity is an improper edge coloring. The linear arboricity of a graph refers to the minimum number that the edge set is divided into linear forests. By using discharging method, we proved a conclusion about a graph G which can be embedded in a surface of non-negative Euler characteristic with maximum degree Δ≥7. If there exist two fixed integers i, j∈{3, 4, 5, 6} such that G has no adjacent i- cycles or j- cycles, then its linear arboricity is $\left\lceil\frac{\Delta}{2}\right\rceil $.

Key words: Euler characteristic, surface, linear arboricity

CLC Number: 

  • O157.5

Fig.1

The forbidden configurations in G"

Fig.2

Distributions of 3-faces where a 7-vetex is incident with three 3-faces"

Fig.3

Distributions of 3-faces where a 8-vetex is incident with four 3-faces"

1 HARARY F . Covering and packing in graphs Ⅰ[J]. Annals of the New York Academy of Sciences, 1970, 175 (1): 198- 205.
doi: 10.1111/j.1749-6632.1970.tb56470.x
2 AKIYAMA J , EXOO G , HARARY F . Covering and packing in graphs Ⅲ: cyclic and acyclic invariants[J]. Mathematica Slovaca, 1980, 30, 405- 417.
3 WU Jianliang . On the linear arboricity of planar graphs[J]. Journal of Graph Theory, 1999, 31 (2): 129- 134.
doi: 10.1002/(SICI)1097-0118(199906)31:2<129::AID-JGT5>3.0.CO;2-A
4 WU Jianliang , WU Yuwen . The linear arboricity of planar graphs of maximum degree seven is four[J]. Journal of Graph Theory, 2008, 58 (3): 210- 220.
doi: 10.1002/jgt.20305
5 CYGAN M , HOU Jianfeng , KOWALIK Ł , et al. A planar linear arboricity conjecture[J]. Journal of Graph Theory, 2011, 69 (4): 403- 425.
6 CHEN Hongyu , TAN Xiang , WU Jianliang , et al. The linear arboricity of planar graphs without 5-, 6-cycles with chords[J]. Graphs and Combinatorics, 2013, 29, 373- 385.
doi: 10.1007/s00373-011-1118-y
7 CHEN Xianglian , WU Jianliang . The linear arboricity of planar graphs without 5-cycles with two chords[J]. Filomat, 2016, 30 (5): 1135- 1142.
doi: 10.2298/FIL1605135C
8 罗朝阳, 孙林. 6-圈至多含一弦平面图的线性荫度[J]. 运筹学学报, 2019, 23 (2): 113- 119.
LUO Zhaoyang , SUN Lin . The linear arboricity of planar graphs with 6-cycles containing at most one chord[J]. Operations Research Transactions, 2019, 23 (2): 113- 119.
9 WANG Huijuan , LIU Bin , WU Jianliang . The linear arboricity of planar graphs without chordal short cycles[J]. Utilitas Mathematica, 2012, 87, 255- 263.
10 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.
doi: 10.1007/s10878-013-9680-2
11 BONDY J A , MURTY U S R . Graph theory with applications[M]. London: Macmillan Press, 1979.
12 WANG Huijuan , WU Jianliang , LIU Bin , et al. On the linear arboricity of graphs embeddable in surfaces[J]. Information Processing Letters, 2014, 114 (9): 475- 479.
doi: 10.1016/j.ipl.2014.03.013
13 陈洪玲, 王慧娟, 高红伟. 可嵌入到欧拉示性数非负的曲面图的线性荫度[J]. 山东大学学报(理学版), 2018, 53 (12): 17- 22.
CHEN Hongling , WANG Huijuan , GAO Hongwei . Linear arboricity of graphs embedded in a surface of non-negative Euler characteristic[J]. Journal of Shandong University(Natural Science), 2018, 53 (12): 17- 22.
14 WU Jianliang , HOU Jianfeng , LIU Guizhen . The linear arboricity of planar graphs with no short cycles[J]. Theoretical Computer Science, 2007, 381, 230- 233.
15 WU Jianliang, HOU Jianfeng, SUN Xiangyong. A note on the linear arboricity of planar graphs without 4-cycles[C]//The Eighth International Symposium on Operations Research and its Applications (ISORA 2009). Beijing, China: World Publishing Corporation, 2009: 174-178.
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