A new neighborhood condition for graphs to be fractional (k, m)-deleted graphs

Let G be a graph of order n, and let k >= 2 and m >= 0 be two integers. 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)l 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 there exists a fractional k-factor G[F(h)] of G with indicator function h such that h(e) = 0 for any e is an element of E(H), where H is any subgraph of G with m edges. In this paper, it is proved that G is a fractional (k, m)-deleted graph if delta(G) >= k 2m, n >= 8k(2) + 4k - 8 + 8m(k + 1) + 4m-2/k+m-1 and vertical bar N(G)(x) U N(G)(y) vertical bar >= n/2 for any two nonadjacent vertices x and y of G such that N(G)(x) boolean AND N(G)(y) not equal (sic). Furthermore, it is shown that the result in this paper is best possible in some sense. ID 2011 Elsevier Ltd. All rights reserved.