1-Factor and Cycle Covers of Cubic Graphs

Let G be a bridgeless cubic graph. Consider a list of k 1-factors of G. Let Ei be the set of edges contained in precisely i members of the k 1-factors. Let k(G) be the smallest |E0| over all lists of k 1-factors of G. Any list of three 1-factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge-covers and for the existence of three 1-factors with empty intersection. Furthermore, if 3(G)0, then 23(G) is an upper bound for the girth of G. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs. Cubic graphs with 4(G)=0 have a 4-cycle cover of length |E(G)| and a 5-cycle double cover. These graphs also satisfy two conjectures of Zhang . We also give a negative answer to a problem stated in .