Delete the edges of a Kneser graph independently of each other with some probability: for what probabilities is the independence number of this random graph equal to the independence number of the Kneser graph itself? We prove a sharp threshold result for this question in certain regimes. Since an independent set in the Kneser graph is the same as a uniform intersecting family, this gives us a ...

It is well known that the automorphism group of the Kneser graph KGn,k is the symmetric group on n letters. For n ≥ 2k + 1, k ≥ 2, we prove that the automorphism group of the stable Kneser graph SGn,k is the dihedral group of order 2n. Let [n] := [1, 2, 3, . . . , n]. For each n ≥ 2k, n, k ∈ {1, 2, 3, . . .}, the Kneser graph KGn,k has as vertices the k-subsets of [n] with edges defined by disj...

Treewidth is an important and well-known graph parameter that measures the complexity of a graph. The Kneser graph Kneser(n, k) is the graph with vertex set ( [n] k ) , such that two vertices are adjacent if they are disjoint. We determine, for large values of n with respect to k, the exact treewidth of the Kneser graph. In the process of doing so, we also prove a strengthening of the Erdős-Ko-...

A Kneser graph KGn,k is a graph whose vertices are all 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 KGn,k is equal to n − 2k + 2. In this paper we discuss the chromatic number of random Kneser graphs and hypergraphs. It was studied in two recent paper...

The Kneser graph K(n, k) has as vertices all k-element subsets of [n] := {1, 2, . . . , n} and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph H(n, k) has as vertices all k-element and (n−k)-element subsets of [n] and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all connected Kneser graphs and bipar...

Let n and k be positive integers. The Kneser graph K n is the graph with vertex set [2n+k] and where two n-subsets A, B ∈ [2n+k] are joined by an edge if A∩B = ∅. In this note we show that the diameter of the Kneser graph K n is equal to d k e+1.

The Kneser graph K (n; k) has as vertices the k-subsets of f1;2;:::;ng. Two vertices are adjacent if the corresponding k-subsets are disjoint. It was recently proved by the rst author 2] that Kneser graphs have Hamilton cycles for n 3k. In this note, we give a short proof for the case when k divides n. x 1. Preliminaries. Suppose that n k 1 are integers and let n] := f1; 2; :::; ng. We denote t...

For natural numbers n, r ∈ N with n ≥ r, the Kneser graph K(n, r) is the graph on the family of r-element subsets of {1, . . . , n} in which two sets are adjacent if and only if they are disjoint. Delete the edges of K(n, r) with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We answer t...

We determine the chromatic number of some graphs flags in buildings type A4, namely Kneser {2,4} vector spaces GF(q)5 for q≥3, and graph {2,3} large q.

We prove that coloring a 3-uniform 2-colorable hypergraph with c colors is NP-hard for any constant c. The best known algorithm [20] colors such a graph using O(n1/5) colors. Our result immediately implies that for any constants k ≥ 3 and c2 > c1 > 1, coloring a k-uniform c1-colorable hypergraph with c2 colors is NP-hard; the case k = 2, however, remains wide open. This is the first hardness re...