a graph $g$ is called a fractional‎ ‎$(k,n',m)$-critical deleted graph if any $n'$ vertices are removed‎ ‎from $g$ the resulting graph is a fractional $(k,m)$-deleted‎ ‎graph‎. ‎in this paper‎, ‎we prove that for integers $kge 2$‎, ‎$n',mge0$‎, ‎$nge8k+n'+4m-7$‎, ‎and $delta(g)ge k+n'+m$‎, ‎if‎ ‎$$|n_{g}(x)cup n_{g}(y)|gefrac{n+n'}{2}$$‎ ‎for each pair of non-adjac...

A graph G is called a fractional (g, f)-deleted graph if G - {e} admits a fractional (g, f)-factor for any e ∈ E(G). A graph G is called a fractional (g, f, n)-critical deleted graph if, after deleting any n vertices from G, the resulting graph is still a fractional (g, f)-deleted graph. The toughness, as the parameter for measuring the vulnerability of communication networks, has received sign...

A graph G is called a fractional (k, n′,m)-critical deleted graph if any n′ vertices are removed from G the resulting graph is a fractional (k,m)-deleted graph. In this paper, we prove that for integers k ≥ 2, n′,m ≥ 0, n ≥ 8k + n′ + 4m− 7, and δ(G) ≥ k + n′ +m, if |NG(x) ∪NG(y)| ≥ n+ n′ 2 for each pair of non-adjacent vertices x, y of G, then G is a fractional (k, n′,m)-critical deleted graph....

A graph G is called a fractional (k, n′,m)-critical deleted graph if any n′ vertices are removed from G the resulting graph is a fractional (k,m)-deleted graph. In this paper, we determine that for integers k ≥ 1, i ≥ 2, n′,m ≥ 0, n > 4ki+ n′ + 4m− 4, and δ(G) ≥ k(i− 1) + n′ + 2m, if |NG(x1) ∪NG(x2) ∪ · · · ∪NG(xi)| ≥ n+ n′ 2 for any independent subset {x1, x2, . . . , xi} of V (G), then G is a...

A graph G is called a P≥k-factor deleted if – e has for any e∈E(G). (P≥k,n)-factor critical every subset V′⊆V(G) with |V′|=n, the G−V′ deleted. Zhou, Bian and Pan [Discrete Appl. Math. (2021) in press] showed that an (n+2)-connected (P≥3,n)-factor its binding number bind(G)>n+32. In this paper, we give new condition graphs, which improves above result.

Let $P(G,lambda)$ be the chromatic polynomial of a graph $G$. A graph $G$ ischromatically unique if for any graph $H$, $P(H, lambda) = P(G,lambda)$ implies $H$ is isomorphic to $G$. In this paper, we determine the chromaticity of all Tur'{a}n graphs with at most three edges deleted. As a by product, we found many families of chromatically unique graphs and chromatic equivalence classes of graph...

We consider graphs G with ∆ = 3 such that χ′(G) = 4 and χ′(G− e) = 3 for every edge e, so-called critical graphs. Jakobsen noted that the Petersen graph with a vertex deleted, P ∗, is such a graph and has average degree only 2 + 2 3 . He showed that every critical graph has average degree at least 2+ 23 , and asked if P ∗ is the only graph where equality holds. We answer his question affirmativ...

the analysis of vulnerability in networks generally involves some questions about how the underlying graph is connected. one is naturally interested in studying the types of disruption in the network that maybe caused by failures of certain links or nodes. in terms of a graph, the concept of connectedness is used in dierent forms to study many of the measures of vulnerability. when certain ver...