On neighborhood condition for graphs to have [a,b]-factors

A Degree Condition For Graphs to Have Connected (g, f)-Factors

A Neighborhood Condition for Graphs to Have [ a , b ]-Factors III

Let a, b, k, and m be positive integers such that 1 ≤ a < b and 2 ≤ k ≤ (b + 1− m)/a. Let G = (V (G), E(G)) be a graph of order |G|. Suppose that |G| > (a + b)(k(a + b − 1) − 1)/b and |NG(x1) ∪ NG(x2) ∪ · · · ∪ NG(xk)| ≥ a|G|/(a+ b) for every independent set {x1, x2, . . . , xk} ⊆ V (G). Then for any subgraph H of G with m edges and δ(G−E(H)) ≥ a, G has an [a, b]-factor F such that E(H) ∩ E(F )...

a degree condition for graphs to have connected (g, f)-factors

On common neighborhood graphs II

Let G be a simple graph with vertex set V (G). The common neighborhood graph or congraph of G, denoted by con(G), is a graph with vertex set V (G), in which two vertices are adjacent if and only if they have at least one common neighbor in G. We compute the congraphs of some composite graphs. Using these results, the congraphs of several special graphs are determined.

A Sufficient Condition for Graphs to Have Hamiltonian [a, b]-Factors

Let a and b be nonnegative integers with 2 ≤ a < b, and let G be a Hamiltonian graph of order n with n ≥ (a+b−4)(a+b−2) b−2 . An [a, b]-factor F of G is called a Hamiltonian [a, b]-factor if F contains a Hamiltonian cycle. In this paper, it is proved that G has a Hamiltonian [a, b]-factor if |NG(X)| > (a−1)n+|X|−1 a+b−3 for every nonempty independent subset X of V (G) and δ(G) > (a−1)n+a+b−4 a+...