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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.

MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
Keywords:
factors; spanning subgraph