Yingzhi Tian

Matching preclusion for n-dimensional torus networks

Abstract The matching preclusion number of a graph is the minimum number of edges whose deletion results in the remaining graph that has neither perfect matchings nor almost perfect matchings. Wang et al. [13] proved that a class of n -dimensional tours networks with even order are super matched. Later, Cheng et al. [8] further showed that all n -dimensional tours networks with even order are super matched. In this paper, we prove that all n -dimensional torus networks with odd order are super matched if n ≥ 3 . Two-dimensional torus networks with odd order is maximally matched except for C 3 □ C 3 . Our results are complementary to those of Wang et al. [13] and Cheng et al. [8] .

The minimum restricted edge-connected graph and the minimum size of graphs with a given edge-degree

Let G=(V(G),E(G)) be a graph. Determining the minimum and/or maximum size (|E(G)|) of graphs with some given parameters is a classic extremal problem in graph theory. For a graph G and e=uv@?E(G), we denote d(e)=d(u)+d(v)-2 the edge-degree of e. In this paper, we obtain a lower bound for the minimum size of graphs with a given order n, a given minimum degree @d and a given minimum edge-degree 2@d+k-2. Moreover, we characterize the extremal graphs for k=0,1,2. As an application, we characterize some kinds of minimum restricted edge connected graphs.

Super R k -vertex-connectedness.

Abstract For a graph G = ( V , E ) , a subset F ⊆ V(G) is called an Rk-vertex-cut of G if G − F is disconnected and each vertex u ∈ V ( G ) − F has at least k neighbours in G − F . The Rk-vertex-connectivity of G, denoted by κk(G), is the cardinality of a minimum Rk-vertex-cut of G. In this paper, we further study the Rk-vertex-connectivity by introducing the concept, called super Rk-vertex-connectedness. The graph G is called super Rk-vertex-connectedness if, for every minimum Rk-vertex-cut S, G − S contains a component which is isomorphic to a certain graph H, where H is related to the graph G and integer k. For the Cayley graphs generated by wheel graphs, H is isomorphic to K2 when k = 1 and H is isomorphic to C4 when k = 2 . In this paper, we show that the Cayley graphs generated by wheel graphs are super R1-vertex-connectedness and super R2-vertex-connectedness. Our studies generalize the main result in [8].

Connectivity of half vertex transitive digraphs

Abstract A bipartite digraph is said to be a half vertex transitive digraph if its automorphism acts transitively on the sets of its bipartition, respectively. In this paper, bipartite double coset digraphs of groups are defined and it is shown that any half vertex transitive digraph is isomorphic to some half double coset digraph, and we show that the connectivity of any strongly connected half transitive digraph is its minimum degree.

Matching preclusion for k-ary n-cubes with odd k≥3

Abstract The matching preclusion number of a graph is the minimum number of edges whose deletion results the remaining graph that has neither perfect matchings nor almost perfect matchings. In this paper, we prove that the matching preclusion number of k -ary n -cubes is 4 n − 1 except k = 3 and n = 2 , where k is odd and k ≥ 3 .

On the existence of super edge-connected graphs with prescribed degrees

Abstract Let G be a connected graph of order n , minimum degree δ ( G ) , and edge-connectivity κ ′ ( G ) . The graph G is maximally edge-connected if κ ′ ( G ) = δ ( G ) and super edge-connected if every minimum edge-cut consists of edges incident with a vertex of minimum degree. A list ( d 1 , … , d n ) is graphic if there is a graph with vertices v 1 , … , v n such that d ( v i ) = d i for 1 ≤ i ≤ n . A graphic list D is super edge-connected if D is the degree list of some super edge-connected graph. We prove that a graphic list D with least element 1 is super edge-connected if and only if (1) ∑ i = 1 n d i ≥ 2 n or (2) ∑ i = 1 n d i = 2 ( n − 1 ) and max { d i : 1 ≤ i ≤ n } = n − 1 . We also give a necessary and sufficient condition for a graphic list with least entry 2 to be super edge-connected, and we show that every graphic list with least element at least 3 is super edge-connected.