Hom complexes of graphs whose codomains are square-free

TL;DR

This paper investigates the topological structure of Hom complexes between graphs, showing that for certain connected graphs, each component is homotopy equivalent to simple spaces, and relates fundamental groups to graph reconfiguration walks.

Contribution

It characterizes the homotopy types of Hom complexes when the codomain graph is square-free, extending understanding of their topological and combinatorial properties.

Findings

Connected components are homotopy equivalent to a point, circle, H, or a double cover of H.

Established a relation between the fundamental group of Hom complexes and reconfiguration walks.

Provided topological classifications for Hom complexes with square-free codomain graphs.

Abstract

The Hom complex $\mathrm{Hom}(G, H)$ of graphs is a simplicial complex associated to a pair of graphs $G$ and $H$, and its homotopy type is of interest in the graph coloring problem and the homomorphism reconfiguration problem. In this paper, we show that if $G$ is a connected graph and $H$ is a square-free connected graph, then every connected component of $\mathrm{Hom}(G, H)$ is homotopy equivalent to a point, a circle, $H$ or a connected double cover over $H$. We also obtain a certain relation between the fundamental group of $\mathrm{Hom}(G,H)$ and realizable walks studied in the homomorphism reconfiguration problem.