On random subgraphs of Kneser and Schrijver graphs [article]

Andrey Borisovich Kupavskii
2015 arXiv   pre-print
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 Lovász 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.
more » ... or a wide range of parameters k = k(n), p = p(n) we show that χ(KG_n,k(p)) is very close to χ(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.
arXiv:1502.00699v2 fatcat:jfjwzkjjebdejoye75qvqedlqe