Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T18:28:48.299Z Has data issue: false hasContentIssue false
  • English
  • Français

On Brooks' Theorem for Sparse Graphs

Published online by Cambridge University Press:  12 September 2008

Jeong Han Kim
Get access Rights & Permissions [Opens in a new window]

Abstract

Let G be a graph with maximum degree Δ(G). In this paper we prove that if the girth g(G) of G is greater than 4 then its chromatic number, χ(G), satisfies

where o(l) goes to zero as Δ(G) goes to infinity. (Our logarithms are base e.) More generally, we prove the same bound for the list-chromatic (or choice) number:

provided g(G) < 4.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 4 , Issue 2 , June 1995 , pp. 97 - 132
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ajtai, M., Komlós, J. and Szemerédi, E. (1980) A note on Ramsey numbers. J. Combinatorial Th. (A) 29 354–360.CrossRefGoogle Scholar
[2]Ajtai, M., Komlós, J. and Szemerédi, E. (1981) A dense infinite Sidon sequence. European J. Combin. 2 1–15.CrossRefGoogle Scholar
[3]Alon, N., Restricted coloring of graphs. Proc. 14th British Combinatorial Conf. to appear.Google Scholar
[4]Bollobás, B. (1978) Chromatic number, girth and maximal degree. Discrete Math. 24 311–314.CrossRefGoogle Scholar
[5]Borodin, O. and Kostochka, A. (1977) On an upper bound of a graph's chromatic number depending on the graph's degree and density. J. Combin. Theory Ser. B 23 247–250.CrossRefGoogle Scholar
[6]Breiman, L. (1968) Probability, Addison-Wesley.Google Scholar
[7]Brooks, R. L. (1941) On colouring the nodes of a network. Proc. Cambridge Philos. Soc. 37 194–197.CrossRefGoogle Scholar
[8]Catlin, P. (1978) A bound on the chromatic number of a graph. Discrete Math. 22 81–83.CrossRefGoogle Scholar
[9]Erdős, P. and Lovász, L. (1974) Problems and results on 3-chromatic hypergraphs and some related questions. Colloq. Math. Soc. Janos Bolyai 10 609–627.Google Scholar
[10]Erdős, P., Rubin, A. L. and Taylor, H. (1979) Choosability in graphs. Proc. West Coast Conf. on Combinatorics, Graph Theory and Computing, Congressus Numerantium 26 125–157.Google Scholar
[11]Frankl, P. and Rödl, V. (1985) Near-perfect coverings in graphs and hypergraphs. Europ. J. Combinatorics 6 317–326.CrossRefGoogle Scholar
[12]Füredi, Z. (1988) Matchings and Covers in Hypergraphs. Graphs and Combinatorics 4 115–206.CrossRefGoogle Scholar
[13]Kahn, J. (1991) Recent results on some not-so-recent hypergraph matching and covering problems. Proc. 1st Int'l Conference on Extremal Problems for Finite Sets, Visegrád, June to appear.Google Scholar
[14]Kahn, J. (1993) Asymptotically good list colorings. J. Combinatorial Theory, Series A. (submitted)Google Scholar
[15]Kahn, J. and Szemerédi, E. (1988) The second eigenvalue of a random regular graph, manuscript, (appeared as first chapter in: Friedman, J., Kahn, J. and Szemerédi, E., On the second eigenvalue of a random regular graph, Proc. 21st STOC, ACM, 1989).Google Scholar
[16]Komlós, J., Pintz, J. and Szemerédi, E. (1982) On Heilbronn's triangle problem. J. London Math. Soc. 25 13–24.CrossRefGoogle Scholar
[17]Kostochka, A.Letter to B. Toft.Google Scholar
[18]Lawrence, J. (1978) Covering the vertex set of a graph with subgraphs of smaller degree. Discrete Math. 21 61–68.CrossRefGoogle Scholar
[19]Milman, V. and Schechtman, G. (1980) Asymptotic Theory of Finite Dimensional Normed Spaces, Springer.Google Scholar
[20]N., Pippenger, unpublished.Google Scholar
[21]Pippenger, N. and Spencer, J. (1989) Asymptotic behavior of the chromatic index for hyper-graphs. J. Combinatorial Th. (A) 51 24–42.CrossRefGoogle Scholar
[22]Rödl, V. (1985) On a packing and covering problem. Europ. J. Combinatorics 5 69–78.CrossRefGoogle Scholar
[23]Shamir, E. and Spencer, J. (1987) Sharp concentration of the chromatic number on random graphs Gn.p. Combinatorica 7 121–129.CrossRefGoogle Scholar
[24]Shearer, J. (1983) A note on the independence number of triangle-free graphs. Discrete Math. 46 83–87.CrossRefGoogle Scholar
[25]Shearer, J. (1991) A note on the independence number of triangle-free graphs, II. J. Combin. Theory Ser. B 53 300–307.CrossRefGoogle Scholar
[26]Spencer, J. (1977) Asymptotic lower bounds for Ramsey functions. Discrete Math. 20 69–76.CrossRefGoogle Scholar
[27]Spencer, J. (1987) Ten Lectures on the Probabilistic Method, Society for Industrial and Applied Mathematics.Google Scholar
[28]Turán, P. (1941) Egy gráfelméleti szélsőértékfeladatról. Mat. Fiz. Lapok 48 436–452; see also: On the theory pf graphs, Colloq. Math. 3 (1954) 19–30.Google Scholar
[29]Vizing, V. G. (1968) Some unsolved problems in graph theory (Russian). Uspehi Mat. Nauk. 23 117–134. English Translation in Russian Math. Surveys 23 125–141.Google Scholar
[30]Vizing, V. G. (1967) Coloring the vertices of a graph in prescribed colors (Russian). Diskret. Analiz. No. 29, Metody Diskret. Anal. v. Teorii Kodov i Shem 101 3–10.Google Scholar