On the chromatic number of almost s-stable Kneser graphs

Abstract

In 2011, Meunier conjectured that for positive integers n, k, r, s with k ≥ 2, r ≥ 2, and n ≥ max({r, s})k, the chromatic number of sstable r-uniform Kneser hypergraphs is equal to n−max({r,s})(k−1) r−1 . It is a strengthened version of the conjecture proposed by Ziegler (2002), andAlon, Drewnowski andLuczak (2009). The problem about the chromatic number of almost s-stable r-uniform Kneser hypergraphs has also been introduced by Meunier (2011).For the r = 2 case of the Meunier conjecture, Jonsson (2012) provided a purely combinatorial proof to confirm the conjecture for s ≥ 4 and n sufficiently large, and by Chen (2015) for even s and any n. The case s = 3 is completely open, even the chromatic number of the usual almost s-stable Kneser graphs.In this paper, we obtain a topological lower bound for the chromatic number of almost s-stable r-uniform Kneser hypergraphs via a different approach. For the case r = 2, we conclude that the chromatic number of almost s-stable Kneser graphs is equal to n − s(k − 1) for all s ≥ 2. Set t = n − s(k − 1). We show that any proper coloring of an almost s-stable Kneser graph must contain a completely multicolored complete bipartite subgraph K ⌈ t 2 ⌉⌊ t 2 ⌋ . It follows that the local chromatic number of almost s-stable Kneser graphs is at least t 2 + 1. It is a strengthened result of Simonyi and Tardos (2007), and Meunier's (2014) lower bound for almost s-stable Kneser graphs.

A study of the neighborhood complex of $-stable Kneser graphs

“…Combining this result with Lovász's topological lower bound on the chromatic number of graphs yielded a new proof about the chromatic number of almost s-stable Kneser graphs SG s∼ m,n , which was determined earlier by Chen [5] in another method.…”

“…Meunier, and the other authors of the papers devoted to the study of the chromatic number of the s-stable Kneser graphs [4,5,8], used combinatorial (Tucker-Ky Fan's lemma and Z p -Tucker lemma), or algebraic tools.…”

On the neighborhood complex of $\vec{s}$-stable Kneser graphs

“…A weaker form of this conjecture has also been known to be true. Actually, in 2017, P. Chen [4] showed that the same formula is valid for almost s-stable Kneser graphs. Similar to Schrijver's proof, his proof was not based on Lovász's bound.…”