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Acyclic edge coloring of planar graphs without a \(3\)-cycle adjacent to a \(6\)-cycle

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Abstract

An acyclic edge coloring of a graph \(G\) is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index \(a'(G)\) of \(G\) is the smallest integer \(k\) such that \(G\) has an acyclic edge coloring using \(k\) colors. Fiamč ik (Math Slovaca 28:139–145, 1978) and later Alon et al. (J Graph Theory 37:157–167, 2001) conjectured that \(a'(G)\le \Delta +2\) for any simple graph \(G\) with maximum degree \(\Delta \). In this paper, we confirm this conjecture for planar graphs without a \(3\)-cycle adjacent to a \(6\)-cycle.

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Acknowledgments

Supported partially by NSFC (No. 11301035) and Health Management and Health Economics Innovation Team, Beijing University of Chinese Medicine. Supported partially by NSFC (No. 11271006).

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Authors and Affiliations

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

    Yiqiao Wang

  2. School of Science, Hangzhou Dianzhi University, Hangzhou , 310018, China

    Qiaojun Shu

  3. School of Mathematics and System Sciences, Shandong University, Jinan, 250100, China

    Jian-Liang Wu & Wenwen Zhang

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  1. Yiqiao Wang
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  2. Qiaojun Shu
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  3. Jian-Liang Wu
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  4. Wenwen Zhang
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Correspondence to Yiqiao Wang.

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Wang, Y., Shu, Q., Wu, JL. et al. Acyclic edge coloring of planar graphs without a \(3\)-cycle adjacent to a \(6\)-cycle. J Comb Optim 28, 692–715 (2014). https://doi.org/10.1007/s10878-014-9765-6

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  • DOI: https://doi.org/10.1007/s10878-014-9765-6

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