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- W2894702993 abstract "Given positive integers $nge 2k$, the 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 Kneser and proved by Lovasz, states that the chromatic number of $KG_{n,k}$ is equal to $n-2k+2$. In this paper, we study the chromatic number of 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 $kge 3$, $chi(KG_{n,k}(1/2)) = n-Theta(sqrt[2k-2]{log_2 n})$, as well as $chi(KG_{n,2}(1/2)) = n-Theta(sqrt[2]{log_2 n cdot log_2log_2 n})$. We also prove that, for any fixed $lge 6$ and $kge Csqrt[2l-3]{log n}$, we have $chi(KG_{n,k}(1/2))ge n-2k+2-2l$, 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." @default.
- W2894702993 created "2018-10-12" @default.
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- W2894702993 date "2018-10-02" @default.
- W2894702993 modified "2023-09-27" @default.
- W2894702993 title "Sharp bounds for the chromatic number of random Kneser graphs" @default.
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