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• W4362474080 abstract "An acyclic edge coloring of a graph is a proper edge coloring in which there are no bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum positive integer $k$ such that $G$ has an acyclic edge coloring with $k$ colors. It has been conjectured by Fiamv{c}'{i}k that $a'(G) le Delta+2$ for any graph $G$ with maximum degree $Delta$. Linear arboricity of a graph $G$, denoted by $la(G)$, is the minimum number of linear forests into which the edges of $G$ can be partitioned. A graph is said to be chordless if no cycle in the graph contains a chord. Every $2$-connected chordless graph is a minimally $2$-connected graph. It was shown by Basavaraju and Chandran that if $G$ is $2$-degenerate, then $a'(G) le Delta+1$. Since chordless graphs are also $2$-degenerate, we have $a'(G) le Delta+1$ for any chordless graph $G$. Machado, de Figueiredo and Trotignon proved that the chromatic index of a chordless graph is $Delta$ when $Delta ge 3$. They also obtained a polynomial time algorithm to color a chordless graph optimally. We improve this result by proving that the acyclic chromatic index of a chordless graph is $Delta$, except when $Delta=2$ and the graph has a cycle, in which case it is $Delta+1$. We also provide the sketch of a polynomial time algorithm for an optimal acyclic edge coloring of a chordless graph. As a byproduct, we also prove that $la(G) = lceil frac{Delta }{2} rceil$, unless $G$ has a cycle with $Delta=2$, in which case $la(G) = lceil frac{Delta+1}{2} rceil = 2$. To obtain the result on acyclic chromatic index, we prove a structural result on chordless graphs which is a refinement of the structure given by Machado, de Figueiredo and Trotignon for this class of graphs. This might be of independent interest." @default.
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• W4362474080 date "2023-08-01" @default.
• W4362474080 modified "2023-09-25" @default.
• W4362474080 title "Acyclic chromatic index of chordless graphs" @default.
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• W4362474080 doi "https://doi.org/10.1016/j.disc.2023.113434" @default.
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