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Odd and even factors with given properties. (English) Zbl 0964.05541
Summary: Let \(G\) be a graph, and let \(g\) and \(f\) be integer-valued functions defined on \(V(G)\) which satisfy \(g(x)\leq f(x)\) and \(g(x)\equiv f(x)\bmod 2\) for all \(x\in V(G)\). Then a spanning subgraph \(F\) of \(G\) is called a \(\{g,g+2, \dots,f\}\)-factor if \(\deg_F(x) \in\{g(x), g(x)+2, \dots, f(x)\}\) for all \(x\in V(G)\). When \(g(x)=1\) for all \(x\in V(G)\), such a factor is called a \((1,f)\)-odd-factor. We give necessary and sufficient conditions for a graph \(G\) to have a \(\{g,g+2,\dots,f\}\)-factor and a \((1,f)\)-odd-factor which contains an arbitrarily given edge of \(G\). From that we derive some interesting results.

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