-factors, complete-factors, and component-deleted subgraphs.

Abstract: In this paper, we consider the relationship between -factors and component-deleted subgraphs of graphs. Let be a graph. A factor of is a complete-factor if every component of is complete. If is a complete-factor of , and is a component of , then is a component-deleted subgraph. Let denote the number of components of . Let be an integer-valued function defined on with even. Enomoto and Tokuda [H. Enomoto, T. Tokuda, Complete-factors and -factors, Discrete Math. 220 (2000) 239–242] proved that if is a complete-factor of with , and has an -factor for each component of , then has an -factor. We extend their result, and show that it suffices to consider a complete-factor of for some specified instead of . Let be a complete-factor of with . If has an -factor for each component of , then has an -factor in each of the following cases: (1) ; (2) is even and ; (3) has no isolated vertices and ; or (4) has no isolated vertices, is even and . We show that the results in this paper are sharp in some sense. [Copyright &y& Elsevier]