Factors and vertex-deleted subgraphs

Abstract

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 2 n -factors in terms of vertex-deleted subgraphs, Ars Combin. 16 (1983) 271–277] proved that for even integer k, if G - x has a k-factor for each x ∈ V ( G ) , 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 G - x has an f-factor for each x ∈ V ( G ) , then G has an f-factor for an integer-valued function f defined on V ( G ) with ∑ x ∈ V ( G ) f ( x ) 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 G - x 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 V ( G ) with ∑ x ∈ V ( G ) f ( x ) even, | V ( G ) - X | ⩾ 2 . We prove that if ∑ x ∈ X deg G ( x ) ⩽ 2 | V ( G ) - X | - 1 and if G - x has an f-factor for each x ∈ V ( G ) - X , then G has an f-factor. Moreover, if G excludes an isolated vertex, then we can replace the condition ∑ x ∈ X deg G ( x ) ≤ 2 | V ( G ) - X | - 1 with ∑ x ∈ X deg G ( x ) ⩽ 2 | V ( G ) - X | + | X | - 3 . Furthermore the condition will be ∑ x ∈ X deg G ( x ) ⩽ 2 | V ( G ) - X | - 1 when | X | = 1 .