On the Hadwiger number of Kneser graphs and their random subgraphs

Abstract

For $n,k\in \mathbb{N}$, let $KG\left(n,k\right)$ be the usual Kneser graph (whose vertices are $k$-sets of $\left\{1,2,\dots ,n\right\}$ with $A\sim B$ if and only if $A\cap B=\varnothing$). The Hadwiger number of a graph $G$, denoted by $h\left(G\right)$, is $max\left\{t:K_t\preccurlyeq G\right\}$, where $H\preccurlyeq G$ if $H$ is a minor of $G$. Previously, lower bounds have been given on the Hadwiger number of a graph in terms of its average degree. In this paper we give lower bounds on $h\left(KG\left(n,k\right)\right)$ and $h\left(KG{\left(n,k\right)}_p\right)$, where $KG{\left(n,k\right)}_p$ is the binomial random subgraph of $KG\left(n,k\right)$ with edge probability $p$. Each of these bounds is larger than previous bounds under certain conditions on $k$ and $p$.