Strongly even cycle decomposable 4-regular line graphs

A graph is even cycle decomposable if its edges can be partitioned into cycles of even length. A graph G is strongly even cycle decomposable if every subdivision of G with an even number of edges is even cycle decomposable. Markström conjectured that for any simple 2-connected cubic graph G, its line graph L(G) is even cycle decomposable. Máčajová and Mazák further asked whether L(G) is strongly even cycle decomposable. In this paper, we resolve this question (as well as Markström’s conjecture) in the affirmative for a class of cubic graphs. We prove that for a (not necessarily simple) 2-connected cubic graph G, if there exists a cycle C in G such that G−V(C) is a linear forest (i.e., a forest whose components are paths), then L(G) is strongly even cycle decomposable. Our main motivation for considering this class of graphs comes from a conjecture of Ash and Jackson that every cyclically 4-edge-connected cubic graph has a dominating cycle (i.e., a cycle whose deletion results in an independent set of vertices). If this conjecture is true, then our result will imply that the line graph of every cyclically 4-edge-connected cubic graph is strongly even cycle decomposable.