A degree condition for fractional ( g , f , n ) -critical covered graphs

: A graph G is called a fractional ( g , f )-covered graph if for any e ∈ E ( G ), G admits a fractional ( g , f )-factor covering e . A graph G is called a fractional ( g , f , n )-critical covered graph if for any W ⊆ V ( G ) with | W | = n , G − W is a fractional ( g , f )-covered graph. In this paper, we demonstrate that a graph G of order p is a fractional ( g , f , n )-critical covered graph if p ≥ ( a + b )( a + b + n + 1) − ( b − m ) n + 2 a + m , δ ( G ) ≥ ( b − m )( b + 1) + 2 a + m + n and for every pair of nonadjacent vertices u and v of G , max { d G ( u ) , d G ( v ) } ≥ ( b − m ) p + ( a + m ) n + 2 a + b , where g and f are integer-valued functions deﬁned on V ( G ) satisfying a ≤ g ( x ) ≤ f ( x ) − m ≤ b − m for every x ∈ V ( G ).


Introduction
All graphs considered here are finite, undirected and simple.Let G be a graph.The vertex set and the edge set of G are denoted by V(G) and E(G), respectively.Let d G (x) denote the degree of a vertex x in G, and N G (x) denote the neighborhood of a vertex x in G. Set N G [x] = N G (x) ∪ {x}.Let X be a vertex subset of G.We use G[X] to denote the subgraph of G induced by X, and write If no two vertices in X are adjacent, then we call X an independent set of G.
For two integer-valued functions g and f with f (x) ≥ g(x) ≥ 0 for any x ∈ V(G), a (g, f )-factor of G is defined as a spanning subgraph F of G such that g(x) ≤ d F (x) ≤ f (x) for any x ∈ V(G).Let E x = {e : e = xy ∈ E(G)}.A fractional (g, f )-indicator function is a function h that assigns each edge of G to a number in [0, 1] so that g(x) ≤ e∈E x h(e) ≤ f (x) for every x ∈ V(G).Let h be a fractional In recent years, the problems related to factors and fractional factors of graphs have raised attention in computer networks and graph theory.Correa and Matamala [1] gave some results about factors of graphs.Li [2] studied [a, b]-factors of K 1,t -free graphs.Zhou, Sun and Xu [3] obtained a result on the existence of edge-disjoint factors in digraphs.Akbari and Kano [4] discussed the existence of factors in r-regular graphs.Li and Cai [5] derived a degree condition for graphs to have [a, b]factors.Zhou et al [6][7][8][9][10][11] gained some results on factors of graphs.Egawa and Kano [12] posed some sufficient conditions for graphs to admit (g, f )-factors.Ota and Tokuda [13] considered the existence of regular factors in K 1,n -free graphs.Liu and Zhang [14] investigated the existence of fractional factors in graphs.Jiang [15,16] discussed fractional factors of graphs.Zhou et al. [17][18][19][20] verified some results on fractional factors of graphs.Yuan and Hao [21] showed a degree condition for a graph to be a fractional [a, b]-covered graph.Zhou, Xu and Sun [22] improved and extended the result, and presented a degree condition for a graph to be a fractional (a, b, n)-critical covered graph.

Theorem 1 ( [22]
).Let a, b and n be integers with n ≥ 0, a ≥ 1 and b ≥ max{2, a}, and let G be a graph of order p with p ≥ (a+b for every pair of nonadjacent vertices u and v of G, then G is a fractional (a, b, n)-critical covered graph.
In this paper, we extend Theorem 1 to fractional (g, f, n)-critical covered graph, and derive the following result.
Theorem 2. Let a, b, m and n be integers satisfying m ≥ 0, n ≥ 0, a ≥ 1 and b ≥ a + m, let G be a graph of order p with p ≥ (a+b)(a+b+n+1)−(b−m)n+2 a+m , and let g and f be integer-valued functions defined on V(G) a+m + n and for every pair of nonadjacent vertices u and v of G, The following result holds if setting m = 0 in Theorem 2.
Corollary 1.Let a, b and n be integers satisfying n ≥ 0 and b ≥ a ≥ 1, let G be a graph of order p with p ≥ (a+b)(a+b+n+1)−bn+2 a , and let g and f be integer-valued functions defined on + n and for every pair of nonadjacent vertices u and v of G, The following result holds if setting n = 0 in Theorem 2.
Corollary 2. Let a, b and m be integers satisfying m ≥ 0, a ≥ 1 and b ≥ a + m, let G be a graph of order p with p ≥ (a+b)(a+b+1)+2 a+m , and let g and f be integer-valued functions defined on a+m and for every pair of nonadjacent vertices u and v of G, then G is a fractional (g, f )-covered graph.

Proof of Theorem 2
The following theorem derived by Li, Yan and Zhang [23] is essential to the proof of Theorem 2.

Theorem 3 ( [23]
).Let G be a graph, and let g and f be integer-valued functions defined on V(G) satisfying 0 ≤ g(x) ≤ f (x) for any x ∈ V(G).Then G is a fractional (g, f )-covered graph if and only if 2, i f S is not independent, 1, i f S is independent and there is an edge joining S and V(G) \ (S ∪ T ), or there is an edge e = uv joining S and T such that d G−S (v) = g(v) f or v ∈ T, 0, otherwise.
We now verify Theorem 2. Let H = G − W for any W ⊆ V(G) with |W| = n.In order to justify Theorem 2, it suffices to show that H is a fractional (g, f )-covered graph.Suppose that H is not a fractional (g, f )-covered graph.Then by Theorem 3, there exists some subset S of V(H) such that where , a contradiction.Therefore, we admit T ∅.Next, we define We may define . Thus, we easily see that x 1 x 2 E(G).According to the hypothesis of Theorem 2 and H = G − W, the following inequalities hold: (2. 3) which is a contradiction.Theorem 2 is proved.In light of Theorem 3, H is not a fractional (g, f )-covered graph, and so G is not a fractional (g, f )critical covered graph.

Conclusions
In this paper, we investigate the relationship between degree conditions and the existence of fractional (g, f, n)-critical covered graphs.A sufficient condition for a graph being a fractional (g, f, n)-critical covered graph is derived.Furthermore, the sharpness of the main result in this paper is illustrated by constructing a special graph class.In addition, some other graph parameter conditions for graphs being fractional (g, f, n)-critical covered graphs can be studied further.