On extremal graphs with at most two internally disjoint Steiner trees connecting any three vertices

Abstract

The problem of determining the smallest number of edges, $h(n;\bar{\kappa }\ge r)$, which guarantees that any graph with $n$ vertices and $h(n;\bar{\kappa }\ge r)$ edges will contain a pair of vertices joined by $r$ internally disjoint paths was posed by Erdös and Gallai. Bollobás considered the problem of determining the largest number of edges $f(n;\bar{\kappa }\le \ell )$ for graphs with $n$ vertices and local connectivity at most $\ell$. One can see that $f(n;\bar{\kappa }\le \ell )=h(n;\bar{\kappa }\ge \ell +1)-1$. These two problems had received a wide attention of many researchers in the last few decades. In the above problems, only pairs of vertices connected by internally disjoint paths are considered. In this paper, we study the number of internally disjoint Steiner trees connecting sets of vertices with cardinality at least 3.

12 pages

Cited by in corpus (7)

• A survey on the generalized connectivity of graphs
• The generalized 3-connectivity of random graphs
• Graphs with large generalized (edge-)connectivity
• On extremal graphs with exactly one Steiner tree connecting any $k$ vertices
• On extremal graphs with at most $\ell$ internally disjoint Steiner trees connecting any n-1 vertices
• Path connectivity of line graphs
• Constructing Internally Disjoint Pendant Steiner Trees in Cartesian Product Networks