Abstract
Let G be a planar graph with maximum degree Δ. It is proved that if Δ ≥ 8 and G is free of k-cycles for some k ∈ {5,6}, then the total chromatic number χ′′(G) of G is Δ + 1.
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References
Behzad, M.: Graphs and their chromatic number, Doctoral Thesis, Michigan State University (1965)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Macmillan Press, London (1976)
Borodin, O.V.: On the total coloring of planar graphs. J. Reine Angew. Math. 394, 180–185 (1989)
Borodin, O.V., Kostochka, A.V., Woodall, D.R.: Total colourings of planar graphs with large maximum degree. J. Graph Theory 26, 53–59 (1997)
Borodin, O.V., Kostochka, A.V., Woodall, D.R.: Total colourings of planar graphs with large girth. Europ. J. Combinatorics 19, 19–24 (1998)
Hou, J., Liu, G., Cai, J.: List edge and list total colorings of planar graphs without 4-cycles. Theoretical Computer Science 369, 250–255 (2006)
Kostochka, A.V.: The total coloring of a multigraph with maximum degree 4. Discrete Math. 17, 161–163 (1977)
Kostochka, A.V.: Upper bounds of chromatic functions on graphs (in Russian), Doctoral Thesis, Novosibirsk 1978
Kostochka, A.V.: The total chromatic number of any multigraph with maximum degree five is at most seven, Discrete Math. 162, 199–214 (1996)
Rosenfeld, M.: On the total coloring of certain graphs. Israel J. Math. 9, 396–402 (1971)
Sanchez-Arroyo, A.: Determining the total coloring number is NP-hard. Discrete Math. 78, 315–319 (1989)
Sanders, D.P., Zhao, Y.: On total 9-coloring planar graphs of maximum degree seven. J. Graph Theory 31, 67–73 (1999)
Vijayaditya, N.: On total chromatic number of a graph. J. London Math. Soc. 3(2), 405–408 (1971)
Vizing, V.G.: Some unsolved problems in graph theory (in Russian), Uspekhi Mat. Nauk 23, 117–134 (1968) English translation in Russian Math. Surveys 23, 125–141
Wang, P., Wu, J.L.: A note on total colorings of planar graphs without 4-cycles. Discussiones Mathematicae Graph Theory 24, 125–135 (2004)
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This work is supported by a research grant NSFC(60673047) and SRFDP(20040422004) of China.
Received: February 27, 2007. Final version received: December 12, 2007.
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Hou, J., Zhu, Y., Liu, G. et al. Total Colorings of Planar Graphs without Small Cycles. Graphs and Combinatorics 24, 91–100 (2008). https://doi.org/10.1007/s00373-008-0778-8
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DOI: https://doi.org/10.1007/s00373-008-0778-8