Acyclic Chromatic Index of Chordless Graphs

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 Fiamč\'ık 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.