On the treewidth of generalized Kneser graphs

TL;DR

This paper determines the treewidth of generalized Kneser graphs for large enough n, providing improved bounds and a precise formula linking parameters, which advances understanding of their structural properties.

Contribution

It establishes the exact treewidth of generalized Kneser graphs for large n, improving previous bounds and clarifying the relationship between parameters.

Findings

Exact treewidth formula for large n

Improved bounds on n relative to k

Characterization of when the formula applies

Abstract

The generalized Kneser graph $K(n,k,t)$ for integers $k>t>0$ and $n>2k-t$ is the graph whose vertices are the $k$-subsets of $\{1,\dots,n\}$ with two vertices adjacent if and only if they share less than $t$ elements. We determine the treewidth of the generalized Kneser graphs $K(n,k,t)$ when $t\ge 2$ and $n$ is sufficiently large compared to $k$. The imposed bound on $n$ is a significant improvement of a previously known bound. One consequence of our result is the following. For each integer $c\ge 1$ there exists a constant $K(c)\ge 2c$ such that $k\ge K(c)$ implies for $t=k-c$ that $$tw(K(n,k,t))=\binom{n}{k}-\binom{n-t}{k-t}-1$$ if and only if $n\ge (t+1)(k+1-t)$ .