On fractional (k, m)-deleted graphs

Abstract

Let G be a graph of order n, and let k ≥ 1 and m ≥ 0 be two integers. In this paper, we introduce firstly the definition of a fractional (k, m)-deleted graph, and show that G is a fractional (k, m)-deleted graph if δ(G) ≥ k + m + (m+1)²-3÷4k n ≥ 4k - 3 + 2(2k + 1)m and max{d_G(.x), d_G(y)} ≥ n÷2 for each pair of nonadjacent vertices x, y of G. This result is best possible in some sense and it is an extension of the result of J. Yu (J. Yu, G. Liu, M. Ma, B. Cao, A degree condition for graphs to have fractional factors, Advances in Mathematics (China) 35(5)(2006), 621-628).