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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 denoted
by , is the minimum positive integer such that has an acyclic
edge coloring with colors. It has been conjectured by Fiam\v{c}\'{\i}k that
for any graph with maximum degree . Linear
arboricity of a graph , denoted by , is the minimum number of linear
forests into which the edges of can be partitioned. A graph is said to be
chordless if no cycle in the graph contains a chord. Every -connected
chordless graph is a minimally -connected graph. It was shown by Basavaraju
and Chandran that if is -degenerate, then . Since
chordless graphs are also -degenerate, we have for any
chordless graph . Machado, de Figueiredo and Trotignon proved that the
chromatic index of a chordless graph is when . 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 , except when and the graph has a cycle, in which
case it is . 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 , unless
has a cycle with , in which case . 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