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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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