Warmth and connectivity of neighborhood complexes of graphs

TL;DR

This paper explores the relationship between the warmth of a graph and the connectivity of its associated Hom complex, providing new bounds and calculations, and advancing understanding of graph coloring and topological graph invariants.

Contribution

It establishes a connection between warmth and Hom complex connectivity, proves a key case of the conjecture relating them, and computes warmth for specific extremal graph families.

Findings

Warmth is bounded by the connectivity of Hom complexes.

Proved that warmth is at most 3 when the Hom complex has infinite first homology.

Calculated warmth for twisted toroidal graphs and graphs excluding certain bipartite subgraphs.

Abstract

In this paper we study a pair of numerical parameters associated to a graph $G$. One the one hand, one can construct $\text{Hom}(K_2, G)$, a space of homomorphisms from a edge $K_2$ into $G$ and study its (topological) connectivity. This approach dates back to the neighborhood complexes introduced by Lov\'asz in his proof of the Kneser conjecture. In another direction Brightwell and Winkler introduced a graph parameter called the warmth $\zeta(G)$ of a graph $G$, based on asymptotic behavior of $d$-branching walks in $G$ and inspired by constructions in statistical physics. Both the warmth of $G$ and the connectivity of $\text{Hom}(K_2,G)$ provide lower bounds on the chromatic number of $G$. Here we seek to relate these two constructions, and in particular we provide evidence for the conjecture that the warmth of a graph $G$ is always less than three plus the connectivity of $\text{Hom}(K_2, G)$. We succeed in establishing a first nontrivial case of the conjecture, by showing that $\zeta(G) \leq 3$ if $\text{Hom}(K_2,G)$ has an infinite first homology group. We also calculate warmth for a family of `twisted toroidal' graphs that are important extremal examples in the context of $\text{Hom}$ complexes. Finally we show that $\zeta(G) \leq n-1$ if a graph $G$ does not have the complete bipartite graph $K_{a,b}$ for $a+b=n$. This provides an analogue for a similar result in the context of $\text{Hom}$ complexes.