Quadrangulations and the Lovász complex
Carmen Arana, Matěj Stehlík
TL;DR
This work characterizes when the Lovász complex $\mathsf{L}(G)$ of a graph is a surface, showing it occurs exactly when $G$ is a non-bipartite quadrangulation of the orbit space $\mathsf{L}(G)/\mathbb{Z}_2$ with every $4$-cycle facial. It then classifies all such Lovász complexes via the ambient surface, and conversely identifies graphs whose Lovász complex is a surface. The authors leverage $\mathbb{Z}_2$-index and cohomological index to obtain strengthened chromatic bounds for quadrangulations, particularly proving that odd quadrangulations of non-orientable surfaces force $\mathrm{ind}_{\mathbb{Z}_2}(\mathsf{L}(G))\ge 2$, and show that $\mathsf{L}(G)$ is non-tidy in these cases. They connect these topological insights to BUT-manifold phenomena and clarify how facial $4$-cycles govern the double-cover and orientability structure of $\mathsf{L}(G)$. Overall, the paper bridges Lovász's topological methods with graph-embedded surface quadrangulations to yield precise classifications and improved index bounds.
Abstract
The Lovász complex $L(G)$ of a graph $G$ is a deformation retract of its neighborhood complex, equipped with a canonical $Z_2$-action. We show that, under mild assumptions, $L(G)$ is homeomorphic to a surface if and only if $G$ is a non-bipartite quadrangulation of the orbit space $L(G)/Z_2$ in which every $4$-cycle is facial. This yields a classification of the Lovász complexes of all such quadrangulations. As an application, we contextualize a result of Archdeacon \emph{et al.}\ and Mohar and Seymour on the chromatic number of quadrangulations, obtaining a stronger statement about the $Z_2$-index.