An Even 2-Factor in the Line Graph of a Cubic Graph

An even 2-factor is one such that each cycle is of even length. A 4- regular graph G is 4-edge-colorable if and only if G has two edge-disjoint even 2- factors whose union contains all edges in G. It is known that the line graph of a cubic graph without 3-edge-coloring is not 4-edge-colorable. Hence, we are interested in whether those graphs have an even 2-factor. Bonisoli and Bonvicini proved that the line graph of a connected cubic graph G with an even number of edges has an even 2-factor, if G has a perfect matching [Even cycles and even 2-factors in the line graph of a simple graph, Electron. J. Combin. 24 (2017), P4.15]. In this paper, we extend this theorem to the line graph of a connected cubic graph G satisfying certain conditions.