Häggkvist–Hell graphs: A class of Kneser-colorable graphs

Abstract

For positive integers n and r we define the Häggkvist–Hell graph, H n : r , to be the graph whose vertices are the ordered pairs ( h , T ) where T is an r -subset of [ n ] , and h is an element of [ n ] not in T . Vertices ( h x , T x ) and ( h y , T y ) are adjacent iff h x ∈ T y , h y ∈ T x , and T x ∩ T y = ∅ . These triangle-free arc transitive graphs are an extension of the idea of Kneser graphs, and there is a natural homomorphism from the Häggkvist–Hell graph, H n : r , to the corresponding Kneser graph, K n : r . Häggkvist and Hell introduced the r = 3 case of these graphs, showing that a cubic graph admits a homomorphism to H 22 : 3 if and only if it is triangle-free. Gallucio, Hell, and Nes˘etr˘il also considered the r = 3 case, proving that H n : 3 can have arbitrarily large chromatic number. In this paper we give the exact values for diameter, girth, and odd girth of all Häggkvist–Hell graphs, and we give bounds for independence, chromatic, and fractional chromatic number. Furthermore, we extend the result of Gallucio et al. to any fixed r ≥ 2 , and we determine the full automorphism group of H n : r , which is isomorphic to the symmetric group on n elements.