Sharp bounds for the chromatic number of random Kneser graphs
Abstract
Given positive integers $n\ge 2k$, a Kneser graph $KG_{n,k}$ is a graph whose vertex set is the collection of all $k$element subsets of the set $\{1,\ldots, n\}$, with edges connecting pairs of disjoint sets. One of the classical results in combinatorics, conjectured by M. Kneser and proved by L. Lovász, states that the chromatic number of $KG_{n,k}$ is equal to $n2k+2$. In this paper, we study the {\it random Kneser graph} $KG_{n,k}(p)$, that is, the graph obtained from $KG_{n,k}$ by including each of the edges of $KG_{n,k}$ independently and with probability $p$. We prove that, for any fixed $k\ge 3$, $\chi(KG_{n,k}(1/2)) = n\Theta(\sqrt[2k2]{\log_2 n})$ and, for $k=2$, $\chi(KG_{n,k}(1/2)) = n\Theta(\sqrt[2]{\log_2 n \cdot \log_2\log_2 n})$. We also prove that, for any fixed $l\ge 6$ and $k\ge C\sqrt[2l3]{\log n}$, we have $\chi(KG_{n,k}(1/2))\ge n2k+22l$, where $C=C(l)$ is an absolute constant. This significantly improves previous results on the subject, obtained by Kupavskii and by Alishahi and Hajiabolhassan. We also discuss an interesting connection to an extremal problem on embeddability of complexes.
 Publication:

arXiv eprints
 Pub Date:
 October 2018
 arXiv:
 arXiv:1810.01161
 Bibcode:
 2018arXiv181001161K
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics