In this work, we study conditions for the existence of length-constrained path-cycle decompositions, that is, partitions of the edge set 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.

Source : oai:arXiv.org:1402.3741

Volume: Vol 19 no. 3

Section: Graph Theory

Published on: October 26, 2017

Submitted on: September 30, 2016

Keywords: Mathematics - Combinatorics,05C38, 05C05, 05C10, 05C75,G.2.2

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