Note on the minimal size of a graph with generalized connectivity kappa_3= 2

TL;DR

This paper refines the minimal size bounds of graphs with three-vertex generalized connectivity 2, providing exact values and constructions for various graph orders.

Contribution

It improves the lower bound for the minimal number of edges and characterizes the minimal size graphs for all orders, including special cases.

Findings

Lower bound improved to e(6/5)n

Existence of graphs attaining the lower bound for all n, except n=9,10

Exact minimal sizes determined for various n

Abstract

The concept of generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$ was introduced by Chartrand et al. in recent years. In our early paper, extremal theory for this graph parameter was started. We determined the minimal number of edges of a graph of order $n$ with $\kappa_{3}= 2$, i.e., for a graph $G$ of order $n$ and size $e(G)$ with $\kappa_{3}(G)= 2$, we proved that $e(G)\geq (6/5)n$, and the lower bound is sharp by constructing a class of graphs, only for $n\equiv 0 \ (mod \ 5)$ and $n\neq 10$. In this paper, we improve the lower bound to $\lceil(6/5)n\rceil$. Moreover, we show that for all $n\geq 4$ but $n= 9, 10$, there always exists a graph of order $n$ with $\kappa_{3}= 2$ whose size attains the lower bound $\lceil(6/5)n\rceil$. Whereas for $n= 9, 10$ we give examples to show that $\lceil(6/5)n\rceil+1$ is the best possible lower bound. This gives a clear picture on the minimal size of a graph of order $n$ with generalized connectivity $\kappa_{3}= 2$.