Hom complexes of graphs whose codomains are square-free

TL;DR

This work classifies the homotopy types of Hom complexes $ ext{Hom}(G,H)$ when the codomain $H$ is square-free, showing each connected component is either contractible, a circle, the graph $H$, or a connected double cover of $H$. The authors construct a universal $2$-covering using $2$-fundamental groups and prove the covering is contractible, which reduces the homotopy type of each component to a fundamental-group calculation tied to realizable $2$-walks. A key exact sequence links $ ext{pi}_1( ext{Hom}(G,H)_f)$ to $ ext{Pi}(f,v)$, the subgroup of realizable elements in $ ext{pi}_1^2(H,f(v))$, and Wrochna’s results yield the precise types in terms of the image of $ ext{pi}_1(G)$ in $ ext{pi}_1(H)$. The approach extends earlier cycle-based results and clarifies the constraints on what topological types can arise as $ ext{Hom}(G,H)$ when $H$ is square-free, with implications for related reconfiguration problems and colorings. The paper thus provides a cohesive framework connecting combinatorial coverings, realizable walks, and $K(\, ext{Pi},1)$ models for a broad class of Hom complexes.

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. %Recently, Soichiro Fujii, Yuni Iwamasa, Kei Kimura, Yuta Nozaki and Akira Suzuki showed that if $G$ is a connected graph and $H$ is a cycle graph then every connected component of $\mathrm{Hom}(G, H)$ is contractible or homotopy equivalent to a circle. 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.

Theorems & Definitions (67)

• Theorem 1.1: Fujii et al. Fujii
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 2.1
• Theorem 2.2: Kozlovbook
• Definition 2.3
• Proposition 2.4: see BM for example
• Remark 2.5