Remarks on fractional (k,m)-deleted graphs

Abstract

Let G be a graph, and k a positive integer. Let h : E(G) → [0, 1] be a function. If Σ_e∋xh(e) = k holds for each x ∈ V(G), then we call G[F_h] a fractional fc-factor of G with indicator function h where F_h = {e ∈ E(G) : h(e) > 0}. A graph G is called a fractional (k,m )-deleted graph if there exists a fractional fc-factor G[F_h] of G with indicator function h such that h(e) = 0 for any e ∈ E(H), where H is any subgraph of G with m edges. In this paper, we present a sufficient condition for the existence of a fractional (k,ℳ )-deleted graph depending on δ(G) and the neighborhood of independent sets. Furthermore, it is shown that the result in this paper is best possible in some sense.