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arXiv · 2510.03698

Quadrangulations and the Lov\'asz complex

Carmen Arana·Matěj Stehlík
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Abstract

The Lov\'asz 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 Lov\'asz 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.

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Carmen Arana, Matěj Stehlík. 2025-10-04. Quadrangulations and the Lov\'asz complex. https://arxiv.org/abs/2510.03698

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