On random subgraphs of Kneser and Schrijver graphs

TL;DR

This paper investigates the chromatic number of random subgraphs of Kneser and Schrijver graphs, showing that it remains close to the original graph's chromatic number under various conditions.

Contribution

It provides bounds on the chromatic number of random subgraphs of Kneser and Schrijver graphs, extending Lovász's classic result to probabilistic settings.

Findings

Chromatic number of random Kneser subgraphs is close to the original with high probability.

Similar bounds are established for Schrijver graphs.

The difference in chromatic number is at most 4 in many cases.

Abstract

A Kneser graph $KG_{n,k}$ is a graph whose vertices are in one-to-one correspondence with $k$-element subsets of $[n],$ with two vertices connected if and only if the corresponding sets do not intersect. A famous result due to Lov\'asz states that the chromatic number of a Kneser graph $KG_{n,k}$ is equal to $n-2k+2$. In this paper we study the chromatic number of a random subgraph of a Kneser graph $KG_{n,k}$ as $n$ grows. A random subgraph $KG_{n,k}(p)$ is obtained by including each edge of $KG_{n,k}$ with probability $p$. For a wide range of parameters $k = k(n), p = p(n)$ we show that $\chi(KG_{n,k}(p))$ is very close to $\chi(KG_{n,k}),$ a.a.s. differing by at most 4 in many cases. Moreover, we obtain the same bounds on the chromatic numbers for the so-called Schrijver graphs, which are known to be vertex-critical induced subgraphs of Kneser graphs.