1-Factor and Cycle Covers of Cubic Graphs

Eckhard Steffen
2014 Journal of Graph Theory  
Let G be a bridgeless cubic graph. Consider a list of k 1-factors of G. Let E_i be the set of edges contained in precisely i members of the k 1-factors. Let μ_k(G) be the smallest |E_0| 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 μ_3(G) = 0, then 2 μ_3(G) is an upper bound
more » ... r 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 μ_4(G) = 0 have a 4-cycle cover of length 4/3 |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 of Zhang.
doi:10.1002/jgt.21798 fatcat:kxcbm7uj2bei7an2dimg3yaafm