A counterexample to a conjecture of Björner and Lovász on the chi-coloring complex

Associated with every graph G of chromatic number χ is another graph G. The vertex set of G consists of all χ-colorings of G, and two χ-colorings are adjacent when they differ on exactly one vertex. According to a conjecture of Björner and Lovász, this graph G must be disconnected. In this note we give a counterexample to this conjecture. One of the most disturbing problems in graph theory is that we only have few methods to prove lower bounds on the chromatic number of graphs. A famous exception is Lovász’s [3] proof of the Kneser conjecture. This paper has indeed introduced a new method into this area and is one of the first applications of topological methods to combinatorics. It shows how to use the Borsuk-Ulam Theorem to derive a (tight) lower bound for the chromatic number of the Kneser graph. Since then, the idea of finding topological obstructions to graph colorings has been extensively studied [2, 4]. In particular, Björner and Lovász made a conjecture generalizing the concept of a topological obstruction to graph coloring (see [1] conjecture 1.6). In this note we provide a counterexample to this general conjecture. To state the general conjecture, we need some definitions. For two graphs G, H, an H-coloring of G is a homomorphism from G to H. Namely, a mapping φ : V (G) → V (H) such that for all edges (x, y) ∈ E(G) we have (φ(x), φ(y)) ∈ E(H). The coloring complex Hom(G,H) is a CW-complex whose 0-cells are the H-colorings of G. The cells of Hom(G,H) are the maps η from the vertices of G to non-empty vertex subsets of H such that η(x) × η(y) ⊆ E(H) for all (x, y) ∈ E(G). The closure of a cell η consists of all cells η̃ such that η̃(v) ⊆ η(v) for all v ∈ V (G). We say that a complex C is k-connected if every map from S to C can be extended to a map from B to C. Equivalently, if all the homotopy groups up to dimension k are trivial. Specifically, (−1)-connected means non-empty, and 0-connected is connected. We denote the chromatic number of a graph G by χ(G). In an attempt to capture some topological obstructions to low chromatic number, Björner and Lovász have made the following conjecture: Conjecture 1. (Björner and Lovász) Let G,H be two graphs such that the coloring complex Hom(G,H) is k-connected. Then χ(H) ≥ χ(G) + k + 1.