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Acyclic edge colorings of planar graphs and seriesparallel graphs

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Abstract

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. Alon et al. conjectured that a′(G) ⩽ Δ(G) + 2 for any graphs. For planar graphs G with girth g(G), we prove that a′(G) ⩽ max{2Δ(G) − 2, Δ(G) + 22} if g(G) ⩾ 3, a′(G) ⩽ Δ(G) + 2 if g(G) ⩾ 5, a′(G) ⩽ Δ(G) + 1 if g(G) ⩾ 7, and a′(G) = Δ(G) if g(G) ⩾ 16 and Δ(G) ⩾ 3. For series-parallel graphs G, we have a′(G) ⩽ Δ(G) + 1.

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Authors and Affiliations

  1. Department of Mathematics, Shandong University, Jinan, 250100, China

    JianFeng Hou, JianLiang Wu, GuiZhen Liu & Bin Liu

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  1. JianFeng Hou
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  2. JianLiang Wu
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  3. GuiZhen Liu
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  4. Bin Liu
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Correspondence to GuiZhen Liu.

Additional information

This work was supported by National Natural Science Foundation of China (Grant No. 10871119) and Natural Science Foundation of Shandong Province (Grant No. Y2008A20).

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Hou, J., Wu, J., Liu, G. et al. Acyclic edge colorings of planar graphs and seriesparallel graphs. Sci. China Ser. A-Math. 52, 605–616 (2009). https://doi.org/10.1007/s11425-008-0124-x

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  • DOI: https://doi.org/10.1007/s11425-008-0124-x

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