A new neighborhood condition for graphs to be fractional -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 ∑ 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, it is proved that G is a fractional ( k , m ) -deleted graph if δ ( G ) ≥ k + 2 m , n ≥ 8 k 2 + 4 k − 8 + 8 m ( k + 1 ) + 4 m − 2 k + m − 1 and ∣ N G ( x ) ∪ N G ( y ) ∣ ≥ n 2 for any two nonadjacent vertices x and y of G such that N G ( x ) ∩ N G ( y ) ≠ 0̸ . Furthermore, it is shown that the result in this paper is best possible in some sense.