On the Multichromatic Number of s‐Stable Kneser Graphs

Abstract

AbstractFor positive integers n and s, a subset [n] is s‐stable if for distinct . The s‐stable r‐uniform Kneser hypergraph is the r‐uniform hypergraph that has the collection of all s‐stable k‐element subsets of [n] as vertex set and whose edges are formed by the r‐tuples of disjoint s‐stable k‐element subsets of [n]. Meunier () conjectured that for positive integers with , and , the chromatic number of s‐stable r ‐uniform Kneser hypergraphs is equal to . It is a generalized version of the conjecture proposed by Alon et al. (). Alon et al. () confirmed Meunier's conjecture for with arbitrary positive integer q. Lin et al. () studied the kth chromatic number of the Mycielskian of the ordinary Kneser graphs for . They conjectured that for . The case was proved by Mycielski (). Lin et al. () confirmed their conjecture for , or when n is a multiple of k or .In this paper, we investigate the multichromatic number of the usual s ‐stable Kneser graphs . With the help of Fan's (1952) combinatorial lemma, we show that Meunier's conjecture is true for r is a power of 2 and s is a multiple of r, and Lin‐Liu‐Zhu's conjecture is true for .