A neighborhood condition for fractional k-deleted graphs

Let k ≥ 3 be an integer, and let G be a graph of order n with n ≥ 9k + 3− 4 √ 2(k − 1)2 + 2. Then a spanning subgraph F of G is called a k-factor if dF (x) = k for each x ∈ V (G). A fractional k-factor is a way of assigning weights to the edges of a graph G (with all weights between 0 and 1) such that for each vertex the sum of the weights of the edges incident with that vertex is k. A graph G is a fractional k-deleted graph if there exists a fractional k-factor after deleting any edge of G. In this paper, it is proved that G is a fractional k-deleted graph if G satisfies δ(G) ≥ k + 1 and |NG(x) ∪ NG(y)| ≥ 12 (n + k − 2) for each pair of nonadjacent vertices x, y of G. Keywords—graph, minimum degree, neighborhood union, fractional k-factor, fractional k-deleted graph.