Neighborhood conditions for the existence of fractional k-factors with prescribed properties

Abstract

Let G be a graph, and fc a positive integer. Let h: E(G) → [0,1] be a function. If Σ_e∋x h(e) = k holds for each x ∈ V(G), then we call G[F_h] a fractional k-factor of G with indicator function h where F_h = {e ∈ E(G): h(e) > 0}. A graph G is called a fractional (fc,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 obtain two new sufficient conditions for graphs to be fractional (fc, m)-deleted graphs. Furthermore, it is shown that the results in this paper are best possible in some sense.