A toughness condition for fractional (k, m)-deleted graphs

Let G be a graph. Let h : E(G) -> [0,1] be a function. If Sigma(e(sic)x) h(e) = k holds for each x is an element of V (G), then we call G[F-h] a fractional k-factor of G with indicator function h where F-h = {e is an element of E(G): h(e) > 0}. A graph G is called a fractional (k, m)-deleted graph if for every e is an element of E(H), there exists a fractional k-factor G[F-h] of G with indicator function h such that h(e) = 0, where H is any subgraph of G with m edges. In this paper, we obtain a toughness condition for a graph to be a fractional (k, m)-deleted graph. This result is best possible in some sense, and it is an extension of Liu's previous result. (c) 2013 Elsevier B.V. All rights reserved.