Critically (k, k)-connected graphs

Abstract

The vertex connectivity of a graph G is denoted by κ(G) and the minimum degree of G is denoted by δ(G). A finite simple graph G is said to be critically (k, k)-connected if $\text{κ(G) = κ(}\text{G}\text{) = k}$ and for each vertex ν of G κ(G − ν) = k − 1 or $\text{κ(}\text{G}\text{− ν) = k −1}$, where $\text{G}$ is the complement of G. The following result is proved: If G is acritically (k, k)-connected graphs, $\text{k ⩾ 2, δ(G) ⩾}\text{1}\text{2}\text{(3k − 1)}$ and $\text{δ(}\text{G}\text{) ⩾}\text{1}\text{2}\text{(3k − 1)}$, then |V(G)|⩽4k. Furthermore, these bounds are sharp for k ⩾ 3.