JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2021, Vol. 56 ›› Issue (11): 71-75.doi: 10.6040/j.issn.1671-9352.0.2020.358

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Total colorings of one type of planar graphs with maximum degree 6

TAN Xiang   

  1. School of Mathematics and Quantitative Economics, Shandong University of Finanace and Economics, Jinan 250014, Shandong, China
  • Published:2021-11-15

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Abstract: Let G be a planar graph with maximum degree Δ≥6 and without 5-cycles, if Δ-vertex isnt incident with 8-cycles, then χ″(G)=Δ+1.

Key words: planar graph, total coloring, cycle

CLC Number: 

  • O157.5
[1] BONDY J A, MURTY U S R. Networks[M] //Graph Theory with Applications. London: Macmillan Education UK, 1976: 191-211.
[2] BEHZAD M. Graphs and their chromatic numbers[D]. Michigan: Michigan State University, 1965.
[3] VIZING V G. Some unsolved problems in graph theory[J]. Russian Mathematical Surveys, 1968, 23(6):125-141.
[4] VIJAYADITYA N. On total chromatic number of a graph[J]. Journal of the London Mathematical Society, 1971,(3):405-408.
[5] ROSENFELD M. On the total coloring of certain graphs[J]. Israel Journal of Mathematics, 1971, 9(3):396-402.
[6] KOSTOCHKA A V. Upper bounds of chromatic functions on graphs[D]. Novosibirsk:[s,L] , 1978.
[7] KOSTOCHKA A V. The total chromatic number of any multigraph with maximum degree five is at most seven[J]. Discrete Mathematics, 1996, 162(1/2/3):199-214.
[8] SANDERS D P, ZHAO Y. On total 9-coloring planar graphs of maximum degree seven[J]. Journal of Graph Theory, 1999, 31(1):67-73.
[9] BORODIN O V, KOSTOCHKA A V, WOODALL D R. Total colorings of planar graphs with large maximum degree[J]. Journal of Graph Theory, 1997, 26(1):53-59.
[10] WANG W F. Total chromatic number of planar graphs with maximum degree ten[J]. Journal of Graph Theory, 2007, 54(2):91-102.
[11] KOWALIK Ł, SERENI J S, ŠKREKOVSKI R. Total-coloring of plane graphs with maximum degree nine[J]. SIAM Journal on Discrete Mathematics, 2008, 22(4):1462-1479.
[12] 蔡华. 平面图的若干染色问题[D]. 济南: 山东大学, 2016. CAI Hua. Some coloring problems of planar graphs[D]. Jinan: Shandong University, 2016.
[13] HOU J F, LIU B, LIU G Z, et al. Total coloring of planar graphs without 6-cycles[J]. Discrete Applied Mathematics, 2011, 159(2/3):157-163.
[14] SHEN L, WANG Y Q. Planar graphs with maximum degree 7 and without 5-cycles are 8-totally-colorable[J]. Discrete Mathematics, 2010, 310(17/18):2372-2379.
[15] SHEN L, WANG Y Q. On the 7 total colorability of planar graphs with maximum degree 6 and without 4-cycles[J]. Graphs and Combinatorics, 2009, 25(3):401-407.
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