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1-factor and cycle covers of cubic graphs. (English) Zbl 1309.05108
Summary: 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 $$\mu_k(G)$$ 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 $$\mu_3(G)\neq0$$, then $$2\mu_3(G)$$ 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 $$\mu_4(G)=0$$ have a 4-cycle cover of length $$\frac{4}{3}|E(G)|$$ and a 5-cycle double cover. These graphs also satisfy two conjectures of Zhang [C.-Q. Zhang, Integer flows and cycle covers of graphs. New York, NY: Marcel Dekker (1996; Zbl 0866.05001)]. We also give a negative answer to a problem stated in [loc. cit.].

##### MSC:
 05C38 Paths and cycles
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