Abstract
A vertex coloring of a graph \(G\) is called acyclic if it is a proper vertex coloring such that every cycle \(C\) receives at least three colors. The acyclic chromatic number of \(G\) is the least number of colors in an acyclic coloring of \(G\). We prove that acyclic chromatic number of any graph \(G\) with maximum degree \(\Delta \ge 4\) and with girth at least \(4\Delta \) is at most \(12\Delta \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alon N, McDiarmid C, Reed B (1991) Acyclic coloring of graphs. Random Struct Algorithms 2:277–288
Bondy JA, Murty USR (1976) Graph theory with applications. Macmillan Press[M], New York
Gerke S, Raemy M (2007) Generalized acyclic edge coloings of graphs with large girth. Discret Math 307:1668–1671
Goncalves D, Montassier M, Pinlou A. Entropy compression method applied to graph coloring, arXiv:1406.4380vl
Greenhill C, Pikhurko O (2005) Bounds on the generalized acyclic chromatic number of bounded degree graphs. Graphs Combin 21:407–419
Grünbaum B (1973) Acyclic colorings of planar graphs. Israel J Math 14:390–408
Molloy M, Reed B (2002) Graph coloring and the probabilistic method, algorithms and combinatorics. Springer, New York
Molloy M, Reed B (1998) Further algorithmic aspects of Loász Local Lemma. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, p 524–529
Acknowledgments
We would like to thank the reviewers for providing some very helpful suggestions for revising this paper. This work is supported by NSFSD (ZR2013AM001, ZR2013AL016) and NSFC (11371355).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cai, J., Feng, B. & Yan, G. Acyclic coloring of graphs with some girth restriction. J Comb Optim 31, 1399–1404 (2016). https://doi.org/10.1007/s10878-015-9829-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-015-9829-2