On Extremal Graphs with at Most $\ell$ Internally Disjoint Steiner Trees Connecting Any $n-1$ Vertices

Abstract

The maximum local connectivity was first introduced by Bollobás. The problem of determining the maximum number of edges in a graph with $\overline{\kappa }\le \ell$ has been studied extensively. We consider a generalization of the above concept and problem. For $S\subseteq V(G)$ and $|S|\ge 2$, the generalized local connectivity $\kappa (S)$ is the maximum number of internally disjoint trees connecting $S$ in $G$. The parameter $\overline{\kappa }_k(G)=\max \{\kappa (S)\,|\,S\subseteq V(G),|S|=k\}$ is called the maximum generalized local connectivity of $G$. In this paper the problem of determining the largest number $f(n;\overline{\kappa }_k\le \ell )$ of edges for graphs of order $n$ that have maximum generalized local connectivity at most $\ell$ is considered. The exact value of $f(n;\overline{\kappa }_k\le \ell )$ for $k=n,n-1$ is determined. For a general $k$, we construct a graph to obtain a sharp lower bound.