On path-cycle decompositions of triangle-free graphs

Andrea Jiménez, Yoshiko Wakabayashi

Published 2014-02-16, updated 2015-06-18Version 2

In this work, we study conditions for the existence of length-constrained path-cycle decompositions, that is, partitions of edge sets of a graph into paths and cycles of a given minimum length. Our main contribution is the characterization of the class of all triangle-free graphs with odd distance at least $3$ that admit a path-cycle decomposition with elements of length at least $4$. As a consequence, it follows that Gallai's conjecture on path decomposition holds in a broad class of sparse graphs.