On the treewidth of generalized q-Kneser graphs

TL;DR

This paper determines the treewidth of generalized q-Kneser graphs for large n, providing improved bounds and explicit formulas, which advances understanding of their structural properties.

Contribution

The paper explicitly calculates the treewidth of generalized q-Kneser graphs for large n, improving previous bounds and offering a precise formula.

Findings

Treewidth of generalized q-Kneser graphs is determined for large n.

The bound on n is significantly improved over previous results.

Explicit formula for the treewidth when n ≥ 3k - t + 9.

Abstract

The generalized $q$-Kneser graph $K_q(n,k,t)$ for integers $k>t>0$ and $n>2k-t$ is the graph whose vertices are the $k$-dimensional subspaces of an $n$-dimensional $F_q$-vectorspace with two vertices $U_1$ and $U_2$ adjacent if and only if $\dim(U_1\cap U_2)<t$. We determine the treewidth of the generalized $q$-Kneser graphs $K_q(n,k,t)$ when $t\ge 2$ and $n$ is sufficiently large compared to $k$. The imposed bound on $n$ is a significant improvement of the previously known bound. One consequence of our results is that the treewidth of each $q$-Kneser graph $K_q(n,k,t)$ with $k>t>0$ and $n\ge 3k-t+9$ is equal to $\gauss{n}{k}-\gauss{n-t}{k-t}-1$.