Homomorphism Complexes and k-Cores

TL;DR

This paper establishes a lower bound on the topological connectivity of graph homomorphism complexes based on a new graph invariant D(G), extending previous theorems and applying the results to analyze phase transitions in random complexes.

Contribution

It introduces the invariant D(G) to generalize connectivity bounds of Hom complexes, extending prior results and exploring phase transitions in random graph complexes.

Findings

Connectivity bound: at least m - D(G) - 2

Generalization of Cukić and Kozlov's theorem

Analysis of homological phase transitions in random complexes

Abstract

We prove that the topological connectivity of a graph homomorphism complex Hom($G,K_m$) is at least $m-D(G)-2$, where $\displaystyle D(G)=\max_{H\subseteq G}\delta(H)$. This is a strong generalization of a theorem of Cuki\'{c} and Kozlov, in which $D(G)$ is replaced by the maximum degree $\Delta(G)$. It also generalizes the graph theoretic bound for chromatic number, $\displaystyle\chi(G)\leq D(G)+1$, as $\displaystyle\chi(G)=\min\{ m:\text{Hom}(G,K_m)\neq\varnothing\}$. Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom$(G(n,p),K_m)$ when $p=c/n$ for a fixed constant $c > 0$.