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Star factors with given properties. (English) Zbl 0723.05099
Summary: A spanning subgraph F of a graph G is called a $$\{K_{1,1},...,K_{1,n}\}$$-factor if each component of F is isomorphic to one of $$\{K_{1,1},...,K_{1,n}\}$$, where $$K_{1,k}$$ denotes the star of order $$k+1$$. Among other results, we show that for a graph G, the following two statements are equivalent:
(1) every edge e of G possesses the property that some $$\{K_{1,1},...,K_{1,n}\}$$-factor of G has a component which is isomorphic to $$K_{1,1}$$ or $$K_{1,2}$$ and contains e;
(2) for all $$S\subset V(G)$$, the number of isolated vertices of G-S is at most $$n| S| -\epsilon (S)$$, where $$n\geq 2$$ and $$\epsilon$$ (S) is defined to be 2n-1 when the subgraph induced by S contains an edge, and to be 0 otherwise.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C05 Trees
##### Keywords:
spanning subgraph; factor; star