On fractional (k,m)-deleted graphs with constrains conditions

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[Fh] a fractional k-factor of G with indicator function h where Fh = {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[Fh] 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 + m + m k+1 , n ≥ 4k + 2k − 6 + (4k 2 +6k−2)m−2 k−1 and max{dG(x), dG(y)} ≥ n 2 for any vertices x and y of G with dG(x, y) = 2. Furthermore, it is shown that the result in this paper is best possible in some sense. Keywords—graph, degree condition, fractional k-factor, fractional (k,m)-deleted graph.