Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-18T04:21:19.079Z Has data issue: false hasContentIssue false
  • English
  • Français

Total chromatic number of graphs of high degree

Part of: Graph theory

Published online by Cambridge University Press:  09 April 2009

H. P. Yap ,
Wang Jian-Fang  and
Zhang Zhongfu
H. P. Yap
Affiliation:
Department of Mathematics, National University of Singapore10 Kent Ridge CrescentSingapore0511
Wang Jian-Fang
Affiliation:
Institute of Applied MathematicsAcademia Sinica Beijing The People'sRepublic of China
Zhang Zhongfu
Affiliation:
Department of Mathematics, Lanzhou Railway InstituteLanzhou, Gansu The People's Republic of China
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using a new proof technique of the first author (by adding a new vertex to a graph and creating a total colouring of the old graph from an edge colouring of the new graph), we prove that the TCC (Total Colouring Conjecture) is true for any graph G of order n having maximum degree at least n - 4. These results together with some earlier results of M. Rosenfeld and N. Vijayaditya (who proved that the TCC is true for graphs having maximum degree at most 3), and A. V. Kostochka (who proved that the TCC is true for graphs having maximum degree 4) confirm that the TCC is true for graphs whose maximum degree is either very small or very big.

MSC classification

Secondary: 05C15: Coloring of graphs and hypergraphs
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 3 , December 1989 , pp. 445 - 452
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Behzad, M., Graphs and Their Chromatic Numbers (Doctoral Thesis, Michigan State Univ., 1965).Google Scholar
[2]Behzad, M., ‘Total concepts in graph theory’, Ars Combin. 23 (1987), 35–40.Google Scholar
[3]Behzad, M., Chartrand, G. and Cooper, J. K., ‘The colour numbers of complete graphs,’ J. London Math. Soc. 42 (1967), 226–228.Google Scholar
[4]Chetwynd, A. G. and Hilton, A. J. W., ‘Regular graphs of high degree are 1-factorizable’, Proc. London Math. Soc. 50 (3) (1985), 193–206.Google Scholar
[5]Kostochka, A. V., ‘The total coloring of a multigraph with maximal degree 4’, Discrete Math. 17 (1977), 161–163.CrossRefGoogle Scholar
[6]Rosenfeld, M., ‘On the total colouring of certain graphs’, Israel J. Math. 9 (3) (1971), 396–402.Google Scholar
[7]Vijayaditya, N., ‘On total chromatic number of a graph’, J. London Math. Soc. (2), 3 (1971), 405–408.CrossRefGoogle Scholar
[8]Vizing, V. G., ‘On an estimate of the chromatic class of a p-graph’ (Russian), Diskret. Analiz 3 (1964), 25–30.Google Scholar
[9]Vizing, V. G., ‘Critical graphs with a given chromatic class’ (Russian), Diskret. Analiz 5 (1965), 9–17.Google Scholar
[10]Vizing, V. G., ‘Some unsolved problems in graph theory’, Uspehi Mat. Nauk 23 (1968), 117–134Google Scholar
(Russian Math. Surveys 23 (1968), 125–142).Google Scholar
[11]Yap, H. P., Some topics in graph theory (Cambridge Univ. Press, 1986).Google Scholar
[12]Yap, H. P., ‘Total colourings of graphs’, Bull. London Math. Soc. 21 (1989), 159–163.Google Scholar