Factors and vertex-deleted subgraphs

A relationship is considered between an f-factor of a graph and that of its vertex-deleted subgraphs. Katerinis [Some results on the existence of image-factors in terms of vertex-deleted subgraphs, Ars Combin. 16 (1983) 271–277] proved that for even integer k, if image has a k-factor for each image, then G has a k-factor. Enomoto and Tokuda [Complete-factors and f-factors, Discrete Math. 220 (2000) 239–242] generalized Katerinis’ result to f-factors, and proved that if image has an f-factor for each image, then G has an f-factor for an integer-valued function f defined on image with image even. In this paper, we consider a similar problem to that of Enomoto and Tokuda, where for several vertices x we do not have to know whether image has an f-factor. Let G be a graph, X be a set of vertices, and let f be an integer-valued function defined on image with image even, image. We prove that if image and if image has an f-factor for each image, then G has an f-factor. Moreover, if G excludes an isolated vertex, then we can replace the condition image with image. Furthermore the condition will be image when image.