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Acyclic Edge Coloring of 1-planar Graphs without 4-cycles

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Abstract

An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles in G. The acyclic chromatic index \(\cal{X}_{\alpha}^{\prime}(G)\) of G is the smallest k such that G has an acyclic edge coloring using k colors. It was conjectured that every simple graph G with maximum degree Δ has \(\cal{X}_{\alpha}^{\prime}(G)\le\Delta+2\). A 1-planar graph is a graph that can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, we show that every 1-planar graph G without 4-cycles has \(\cal{X}_{\alpha}^{\prime}(G)\le\Delta+22\).

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Wei-fan Wang

  2. School of Management, Beijing University of Chinese Medicine, Beijing, 100029, China

    Yi-qiao Wang

  3. School of Mathematics and Information Science, Weifang University, Weifang, 261061, China

    Wan-shun Yang

Authors
  1. Wei-fan Wang
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  2. Yi-qiao Wang
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  3. Wan-shun Yang
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Correspondence to Wan-shun Yang.

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The authors declare no conflict of interest.

Additional information

Research supported by the National Natural Science Foundation of China (No.12031018).

Research supported by the National Natural Science Foundation of China (No.12071048) and Science and Technology Commission of Shanghai Municipality (No.18dz2271000).

Research supported by the National Natural Science Foundation of China (No.12071351), Doctoral Scientific Research Foundation of Weifang University (No.2021BS01) and Natural Science Foundation of Shandong Province (No.ZR2022MA060).

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Wang, Wf., Wang, Yq. & Yang, Ws. Acyclic Edge Coloring of 1-planar Graphs without 4-cycles. Acta Math. Appl. Sin. Engl. Ser. 40, 35–44 (2024). https://doi.org/10.1007/s10255-024-1101-z

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  • DOI: https://doi.org/10.1007/s10255-024-1101-z

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