1‐Factor and Cycle Covers of Cubic Graphs. Issue 3 (14th April 2014) - Steffen, Eckhard
Abstract 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 μk(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 μ3(G)≠0, then 2μ3(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 μ4(G)=0 have a 4‐cycle cover of length 43|E(G)| and a 5‐cycle double cover. These graphs also satisfy two conjectures of Zhang 18. We also give a negative answer to a problem stated in 18.