Haggkvist-Hell Graphs: A class of Kneser-colorable graphs

TL;DR

This paper introduces and analyzes Haggkvist-Hell graphs, a new class of Kneser-colorable graphs, providing exact parameters and extending known results to all r ≥ 2, with implications for graph homomorphisms and automorphisms.

Contribution

It defines Haggkvist-Hell graphs, computes their diameter, girth, and automorphism group, and extends previous results on chromatic numbers to all r ≥ 2.

Findings

Exact diameter, girth, and odd girth for all Haggkvist-Hell graphs.

Bounds for independence, chromatic, and fractional chromatic numbers.

Automorphism group is isomorphic to the symmetric group on n elements.

Abstract

For positive integers n and r we define the Haggkvist-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 \in T_y, h_y \in T_x, and T_x and T_y are disjoint. These triangle-free arc transitive graphs are an extension of the idea of Kneser graphs, and there is a natural homomorphism from the Haggkvist-Hell graph, H_{n:r}, to the corresponding Kneser graph, K_{n:r}. Haggkvist 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 Nesetril 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 Haggkvist-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 \ge 2, and we determine the full automorphism group of H_{n:r}, which is isomorphic to the symmetric group on n elements.