On even cycle decompositions of line graphs of cubic graphs

An even cycle decomposition of a graph is a partition of its edges into cycles of even length. In 2012, Markstrom conjectured that the line graph of every 2-connected cubic graph has an even cycle decomposition and proved this conjecture for cubic graphs with oddness at most 2. However, for 2-connected cubic graphs with oddness 2, Markstrom only considered these graphs with a chordless 2-factor. (A chordless 2-factor of a graph is a 2-factor consisting of only induced cycles.) In this paper, we first construct an infinite family of 2-connected cubic graphs with oddness 2 and without chordless 2-factors. We then give a complete proof of Markstrom's result and further prove this conjecture for cubic graphs with oddness 4. (C) 2020 Elsevier B.V. All rights reserved.