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Complete-factors and (\(g,f\))-factors. (English) Zbl 1017.05088

Summary: A factor \(F\) of a graph is called a complete-factor if each component of \(F\) is complete. Let \(G\) be a graph, \(F\) be a complete-factor of \(G\) with \(\omega(F)\geq 2\) and \(f\), \(g\) be two integer-valued functions defined on \(V(G)\) with \(f(x)\geq g(x)\) for all \(x\in V(G)\). It is proved that if \(\omega(F)\equiv 0\pmod 2\), or \(f(V(G))\) even and \(f(x)\equiv g(x)\pmod 2\) for all \(x\in V(G)\), and if \(G- V(C)\) has a \((g,f)\)-factor for each component \(C\) of \(F\), then \(G\) has a \((g,f)\)-factor. We show that the results in this paper are best possible.

MSC:

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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References:

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