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On the Entire Coloring Conjecture

Published online by Cambridge University Press:  20 November 2018

Daniel P. Sanders  and
Yue Zhao
Daniel P. Sanders
Affiliation:
23 Cliff Road Belle Terre, NY 11777 USA, email: [email protected]
Yue Zhao
Affiliation:
Department of Mathematics University of Central Florida PO Box 161364 Orlando, FL 32816-1364 USA, email: [email protected]
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Abstract

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The Four Color Theorem says that the faces (or vertices) of a plane graph may be colored with four colors. Vizing’s Theorem says that the edges of a graph with maximum degree $\Delta$ may be colored with $\Delta \,+\,1$ colors. In 1972, Kronk and Mitchem conjectured that the vertices, edges, and faces of a plane graph may be simultaneously colored with $\Delta \,+\,4$ colors. In this article, we give a simple proof that the conjecture is true if $\Delta \,\ge \,6$.

Keywords

05C1505C10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 43 , Issue 1 , 01 March 2000 , pp. 108 - 114
Copyright
Copyright © Canadian Mathematical Society 2000

References

[1] Appel, K. and Haken, W., Every planar map is four colorable. Bull. Amer. Math. Soc. 82 (1976), 711–712.Google Scholar
[2] Borodin, O. V., Solution of Ringel's problem on vertex-face coloring of plane graphs and coloring of 1-planar graphs (Russian). Metody Diskret. Analiz. 41 (1984), 12–26.Google Scholar
[3] Borodin, O. V., On the total coloring of planar graphs. J. Reine Angew. Math. 394 (1989), 180–185.Google Scholar
[4] Borodin, O. V., Structural theorem on plane graphs with application to the entire coloring number. J. Graph Theory 23 (1996), 233–239.Google Scholar
[5] Fiamčík, J., Simultaneous colouring of 4-valent maps. Mat. Čas. 21 (1971), 9–13.Google Scholar
[6] Jensen, T. R. and B. Toft, Graph Coloring Problems. John Wiley & Sons, Inc., New York, 1995, 47–48, 86–89, 193–194.Google Scholar
[7] Jucovič, E., On a problem in map colouring. Mat. Čas. 19 (1969), 225–227.Google Scholar
[8] Kronk, H. V. and Mitchem, J., The entire chromatic number of a normal graph is at most seven. Bull. Amer. Math. Soc. 78 (1972), 799–800.Google Scholar
[9] Kronk, H. V. and Mitchem, J., A seven-color theorem on the sphere. Discrete Math. 5 (1973), 253–260.Google Scholar
[10] Melnikov, L. S., Problem 9. Recent Advances in Graph Theory (ed. M. Fiedler), Academia Praha, Prague, 1975, 543.Google Scholar
[11] Molloy, M. and Reed, B., A bound on the total chromatic number. Combinatorica 18 (1998), 241–280.Google Scholar
[12] Ringel, G., Ein Sechsfarbenproblem auf der Kugel. Abh. Math. Sem. Univ. Hamburg 29 (1965), 107–117.Google Scholar
[13] Robertson, N., Sanders, D. P., Seymour, P. D. and Thomas, R., The four-colour theorem. J. Combin. Theory Ser. B 70 (1997), 2–44.Google Scholar
[14] Sanders, D. P. and Zhao, Y., On simultaneous edge-face colorings of plane graphs. Combinatorica 17 (1997), 441–445.Google Scholar
[15] Sanders, D. P. and Zhao, Y., On total 9-coloring planar graphs of maximum degree seven. J. Graph Theory 31 (1999), 67–73.Google Scholar
[16] Sanders, D. P. and Zhao, Y., Planar graphs of maximum degree seven are Class I. Submitted.Google Scholar
[17] Vizing, V. G., On an estimate of the chromatic index of a p-graph (Russian).Metody Diskret. Analiz. 3 (1964), 25–30.Google Scholar
[18] Vizing, V. G., Critical graphs with given chromatic index (Russian). Metody Diskret. Analiz. 5 (1965), 9–17.Google Scholar
[19] Waller, A. O., Simultaneously colouring the edges and faces of plane graphs. J.Combin. Theory Ser. B 69 (1997), 219–221.Google Scholar