A sufficient condition for graphs to be fractional -deleted graphs

Let G be a graph, and k a positive integer. Let h : E ( G ) → [ 0 , 1 ] be a function. If ∑ 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 ( 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 ∈ E ( H ) , where H is any subgraph of G with m edges. In this paper, we use a binding number to obtain a sufficient 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 Zhou’s previous results.