Graphs with large generalized 3-connectivity

TL;DR

This paper investigates the generalized 3-connectivity of graphs, providing bounds, characterizations of extremal cases, and advancing understanding of how many internally disjoint trees connect any three vertices.

Contribution

It establishes sharp bounds for the generalized 3-connectivity and characterizes graphs that attain the maximum and near-maximum values.

Findings

Bounds of 1 and n-2 for -connectivity are proven.

Graphs with -connectivity of n-2 and n-3 are characterized.

Theoretical framework for analyzing generalized connectivity in graphs.

Abstract

Let $S$ be a nonempty set of vertices of a connected graph $G$. A collection $T_1,..., T_\ell$ of trees in $G$ is said to be internally disjoint trees connecting $S$ if $E(T_i)\cap E(T_j)= \emptyset$ and $V(T_i)\cap V(T_j)=S$ for any pair of distinct integers $i, j$, where $1 \leq i, j \leq r$. For an integer $k$ with $2 \leq k \leq n$, the generalized $k$-connectivity $\kappa_k(G)$ of $G$ is the greatest positive integer $r$ such that $G$ contains at least $r$ internally disjoint trees connecting $S$ for any set $S$ of $k$ vertices of $G$. Obviously, $\kappa_2(G)$ is the connectivity of $G$. In this paper, sharp upper and lower bounds of $\kappa_3(G)$ are given for a connected graph $G$ of order $n$, that is, $1 \leq \kappa_3(G) \leq n - 2$. Graphs of order $n$ such that $\kappa_3(G) = n - 2, n - 3$ are characterized, respectively.