[Paper Review] The minimal size of a graph with generalized connectivity $κ_3 = 2$

Shasha Li, Xueliang Li+1 more|arXiv (Cornell University)|Jan 20, 2011

Interconnection Networks and Systems1 references15 citations

TL;DR

This paper establishes the minimum number of edges required for a graph to achieve generalized 3-connectivity $\kappa_3 = 2$, proving that any such graph of order $n$ must have at least $\frac{6}{5}n$ edges. The authors construct a family of graphs achieving this bound, demonstrating it is tight and providing a sharp lower bound for $\kappa_3 = 2$ graphs.

ABSTRACT

Let $G$ be a nontrivial connected graph of order $n$ and $k$ an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $κ(S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$ such that $V(T_i)\cap V(T_j)=S$ for every pair $i,j$ of distinct integers with $1\leq i,j\leq \ell$. Chartrand et al. generalized the concept of connectivity as follows: The $k$-$connectivity$, denoted by $κ_k(G)$, of $G$ is defined by $κ_k(G)=$min$\{κ(S)\}$, where the minimum is taken over all $k$-subsets $S$ of $V(G)$. Thus $κ_2(G)=κ(G)$, where $κ(G)$ is the connectivity of $G$. This paper mainly focuses on the minimal number of edges of a graph $G$ with $κ_{3}(G)= 2$. For a graph $G$ of order $v(G)$ and size $e(G)$ with $κ_{3}(G)= 2$, we obtain that $e(G)\geq 6/5v(G)$, and the lower bound is sharp by showing a class of examples attaining the lower bound.

Motivation & Objective

• To determine the minimal number of edges in a graph with generalized 3-connectivity $\kappa_3 = 2$.
• To establish a tight lower bound on the size of such graphs in terms of their order.
• To construct explicit families of graphs that achieve the theoretical minimum edge count.
• To characterize the structural conditions under which the lower bound is attained.
• To resolve the discrepancy between edge count and $\kappa_3$ value by identifying cases where $\frac{6}{5}n$ edges do not suffice (e.g., order 10 graphs).

Proposed method

• Define $\kappa_3(G)$ as the minimum, over all 3-vertex subsets $S$, of the maximum number of edge-disjoint trees connecting $S$ with only $S$ in common.
• Use degree constraints: since $\kappa_3(G) = 2$, the minimum degree $\delta(G) \geq 2$, and vertices of degree 2 form a stable set.
• Partition vertices into $X$ (degree 2) and $Y$ (degree $\geq 3$), and derive edge count via $e(G) = 2|X| + m'$, where $m'$ is the number of edges within $Y$.
• Apply the inequality $5|X| + 2m' \geq 3n$ from degree sum constraints to derive $e(G) \geq \frac{6}{5}n$.
• Construct a family of graphs $H_k$ with $5k$ vertices and $6k$ edges ($e(H_k) = \frac{6}{5}v(H_k)$) to show the bound is tight for $k \neq 2$.
• Use case analysis on vertex subsets $S$ of size 3 to verify $\kappa_3(H_k) = 2$ via explicit construction of two internally disjoint trees for each $S$.

Experimental results

Research questions

• RQ1 What is the minimal number of edges a graph can have while maintaining $\kappa_3(G) = 2$?
• RQ2 Can the lower bound $e(G) \geq \frac{6}{5}v(G)$ be achieved, and if so, under what structural conditions?
• RQ3 Why does the bound fail for graphs of order 10 with 12 edges, despite satisfying $e(G) = \frac{6}{5}v(G)$?
• RQ4 What are the necessary and sufficient conditions for equality in the edge bound for $\kappa_3(G) = 2$?
• RQ5 How do the degree distribution and vertex partitioning affect the existence of internally disjoint trees for 3-sets?

Key findings

• The minimal size of a graph with $\kappa_3(G) = 2$ is $e(G) \geq \frac{6}{5}v(G)$, and this bound is sharp.
• Equality holds if and only if the graph has exactly $\frac{3}{5}v(G)$ vertices of degree 2 forming a stable set, all other vertices have degree 3, and there are no edges within the high-degree set.
• A family of graphs $H_k$ with $5k$ vertices and $6k$ edges achieves the bound for all $k \neq 2$, proving its tightness.
• For $k = 2$, the graph $H_2$ has 10 vertices and 12 edges but satisfies $\kappa_3(H_2) = 1$, showing that $\frac{6}{5}n$ edges do not guarantee $\kappa_3 = 2$.
• The construction of two internally disjoint trees for every 3-vertex subset $S$ confirms $\kappa_3(H_k) = 2$ when $k \neq 2$.
• The bound is not achievable for graphs of order 10, as shown by the counterexample of any connected graph on 10 vertices with 12 edges having $\kappa_3 = 1$.