Colouring normal quadrangulations of projective spaces

TL;DR

This work investigates how topological constraints govern the chromatic number of normal quadrangulations of real projective spaces. It provides a constructive negative result in dimension three—normal quadrangulations of $\mathbb{R}\mathrm{P}^{3}$ can have arbitrarily large chromatic numbers—while establishing a general obstruction that prevents any normal quadrangulation of $\mathbb{R}\mathrm{P}^{d}$ from being $3$-colorable for all $d\ge2$. The authors introduce a cohomology/intersection-theory framework to derive higher-dimensional 3-colorability obstructions and apply it to $\mathbb{R}\mathrm{P}^{d}$ and even-order lens spaces, alongside an alternative proof of Youngs' theorem in $\mathbb{R}\mathrm{P}^{2}$. Collectively, the results reveal stark contrasts between colorability phenomena on nonorientable versus orientable settings and extend the toolkit for analyzing KS- and normal-quadrangulations in higher dimensions.

Abstract

Youngs proved that every non-bipartite quadrangulation of the projective plane $\mathbb{R}\mathrm{P}^2$ is 4-chromatic. Kaiser and Stehlík [J. Combin. Theory Ser. B 113 (2015), 1-17] generalised the notion of a quadrangulation to higher dimensions and extended Youngs' theorem by proving that every non-bipartite quadrangulation of the $d$-dimensional projective space $\mathbb{R}\mathrm{P}^d$ with $d \geq 2$ has chromatic number at least $d+2$. On the other hand, Hachimori et al. [European. J. Combin. 125 (2025), 104089] defined another kind of high-dimensional quadrangulation, called a normal quadrangulation. They proved that if a non-bipartite normal quadrangulation $G$ of $\mathbb{R}\mathrm{P}^d$ with any $d \geq 2$ satisfies a certain geometric condition, then $G$ is $4$-chromatic, and asked whether the geometric condition can be removed from the result. In this paper, we give a negative solution to their problem for the case $d=3$, proving that there exist 3-dimensional normal quadrangulations of $\mathbb{R}\mathrm{P}^3$ whose chromatic number is arbitrarily large. Moreover, we prove that no normal quadrangulation of $\mathbb{R}\mathrm{P}^d$ with any $d \geq 2$ has chromatic number $3$.

Figures (6)

• Figure 1: The local structure of a $3$-dimensional KS-quadrangulation. Left: A portion of the underlying simplicial complex. Right: A part of the KS-quadrangulation (in red).
• Figure 2: Left: A triangular region $R$ in the proof of Theorem \ref{['t:complete']}. Right: and segments $\boldsymbol{p}\boldsymbol{t}_i$ and $\boldsymbol{q}\boldsymbol{t}_i$, $i=0,\dots,k$.
• Figure 3: Left: The ball $B_0$ in the proof of Theorem \ref{['t:complete']} with $k=2$. Right: The even-sided graph $H$.
• Figure 4: Annuli $A_1$ and $A_2$ in the proof of Theorem \ref{['t:complete']}.
• Figure 5: Put curves in each face joining midpoints of the edges of $E_t$.

Theorems & Definitions (20)

• Theorem 1.1: Youngs, 1996
• Theorem 1.2: Kaiser and Stehlı́k, 2015
• Theorem 1.3: Hachimori, Nakamoto, and Ozeki, 2025
• Theorem 1.5
• Theorem 1.7
• Theorem 2.1: Erickson, 2014
• proof : Proof of Theorem \ref{['t:complete']}
• proof : Alternative proof of Theorem \ref{['t-young']}
• Theorem 4.1
• Remark 4.2