Abstract
A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by x′ a (G), is the least number of colors such that G has an acyclic edge k-coloring. Let G be a graph with maximum degree Δ and girth g(G), and let 1 ⩽ r ⩽ 2Δ be an integer. In this paper, it is shown that there exists a constant c > 0 such that if \(g(G) \geqslant \frac{{c\Delta }} {r}\log \left( {\Delta ^2 /r} \right)\) then x′ a (G) ⩽ Δ+r + 1, which generalizes the result of Alon et al. in 2001. When G is restricted to series-parallel graphs, it is proved that x′ a (G) = Δ if Δ ⩾ 4 and g(G) ⩾ 4; or Δ ⩾ 3 and g(G) ⩾ 5.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alon N. A parallel algorithmic version of the local lemma. Random Struct Algor, 1991, 2: 367–378
Alon N, Spencer J. The Probabilistic Method. New York: Wiley, 1992
Alon N, Sudakov B, Zaks A. Acyclic edge colorings of graphs. J Graph Theory, 2001, 37: 157–167
Basavaraju M, Chandran L S. Acyclic edge coloring of subcubic graphs. Discrete Math, 2008, 308: 6650–6653
Basavaraju M, Chandran L S. Acyclic edge coloring of graphs with maximum degree 4. J Graph Theory, 2009, 61: 192–209
Coleman T F, Cai J. The acyclic coloring problem and estimation of spare Hession matrices. SIAM J Algebr Discrete Math, 1986, 7: 221–235
Coleman T F, Moré J J. Estimation of spare Hession matrices and graph coloring problems. Math Prog, 1984, 28: 243–270
Duffin R J. Topology of series-parallel networks. J Math Anal Appl, 1965, 10: 303–318
Erdős P, Lovász L. Problems and results on 3-chromatic hypergraph and some related questions. In: Hajnal A, et al., eds. Infinite and Finite Sets. Amsterdam: North-Holland, 1975
Fertin G, Raspaud A, Reed B. Star coloring of graphs. J Graph Theory, 2004, 47: 163–182
Hou J, Wu J, Liu G, et al. Acyclic edge colorings of planar graphs and seriell-parallel graphs. Sci China Ser A, 2009, 52: 605–616
Hou J, Wu J, Liu G, et al. Acyclic edge chromatic number of outerplanar graphs. J Graph Theory, 2010, 64: 22–36
Hou J, Liu G, Wu J. Acyclic edge coloring of planar graphs without small cycles. Graphs Combin, 2012, 28: 215–226
Kostochka A, Sopena E, Zhu X. Acyclic and oriented chromatic numbers of graphs. J Graph Theory, 1997, 24: 331–340
Molloy M, Reed B. Further algorithmic aspects of the local lemma. In: Proceedings of the 30th Annual ACM Symposium in Theory of Computing. New York: ACM Press, 1998, 524–529
Muthu R, Narayanan N, Subramanian C R. Acyclic edge colouring of outerplanar graphs. Lect Notes Comput Sci, 2007, 4508: 144–152
Skulrattanakulchai S. Acyclic colorings of subcubic graphs. Inform Processing Lett, 2004, 92: 161–167
Spencer J. Asymptotic lower bounds for Ramsey functions. Discrete Math, 1977, 20: 69–76
Vizing V G. On an estimate of the chromatic class of a p-graph (in Russian). Metody Diskret Anal, 1964, 3: 25–30
Wu J, Wu Y. The entire coloring of series-parallel graphs. Acta Math Appl Sin, 2005, 21: 61–66
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lin, Q., Hou, J. & Liu, Y. Acyclic edge coloring of graphs with large girths. Sci. China Math. 55, 2593–2600 (2012). https://doi.org/10.1007/s11425-012-4442-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-012-4442-7