Acyclic edge coloring of planar graphs without a $$3$$ 3 -cycle adjacent to a $$6$$ 6 -cycle

Yiqiao Wang (), Qiaojun Shu, Jian-Liang Wu and Wenwen Zhang
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Yiqiao Wang: Beijing University of Chinese Medicine
Qiaojun Shu: Hangzhou Dianzhi University
Jian-Liang Wu: Shandong University
Wenwen Zhang: Shandong University

Journal of Combinatorial Optimization, 2014, vol. 28, issue 3, No 14, 692-715

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

Keywords: Acyclic edge coloring; Planar graph; Cycles; Maximum degree (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10878-014-9765-6

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