On the difference of two generalized connectivities of a graph

Abstract

The concept of k-connectivity $\kappa '_{k}(G)$ of a graph G, introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. Another generalized connectivity of a graph G, named the generalized k-connectivity $\kappa _{k}(G)$, mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of these two parameters by showing that for a connected graph G of order n, if $\kappa '_k(G)\ne n-k+1$ where $k\ge 3$, then $0\le \kappa '_k(G)-\kappa _k(G)\le n-k-1$; otherwise, $-\lfloor \frac{k}{2}\rfloor +1\le \kappa '_k(G)-\kappa _k(G)\le n-k$. Moreover, all of these bounds are sharp. Some specific study is focused for the case $k=3$. As results, we characterize the graphs with $\kappa '_3(G)=\kappa _3(G)=t$ for $t\in \{1, n-3, n-2\}$, and give a necessary condition for $\kappa '_3(G)=\kappa _3(G)$ by showing that for a connected graph G of order n and size m, if $\kappa '_3(G)=\kappa _3(G)=t$ where $1\le t\le n-3$, then $m\le {n-2\atopwithdelims ()2}+2t$. Moreover, the unique extremal graph is given for the equality to hold.