The width of quadrangulations of the projective plane

TL;DR

This paper investigates the properties of quadrangulations of the projective plane, establishing bounds on edge-width, face-width, and odd cycle transversals, with implications for graph coloring and topological graph theory.

Contribution

It introduces new bounds on edge-width and face-width for non-bipartite quadrangulations of the projective plane, and provides a novel bound on odd cycle transversals, partially answering open questions.

Findings

Edge-width at most (1+√(8n-7))/2, sharp for infinitely many n.

Face-width at most (1+√(16n-15))/4, close to optimal.

Existence of small odd cycle transversals inducing a single edge.

Abstract

We show that every $4$-chromatic graph on $n$ vertices, with no two vertex-disjoint odd cycles, has an odd cycle of length at most $\tfrac12\,(1+\sqrt{8n-7})$. Let $G$ be a non-bipartite quadrangulation of the projective plane on $n$ vertices. Our result immediately implies that $G$ has edge-width at most $\tfrac12\,(1+\sqrt{8n-7})$, which is sharp for infinitely many values of $n$. We also show that $G$ has face-width (equivalently, contains an odd cycle transversal of cardinality) at most $\tfrac14(1+\sqrt{16 n-15})$, which is a constant away from the optimal; we prove a lower bound of $\sqrt{n}$. Finally, we show that $G$ has an odd cycle transversal of size at most $\sqrt{2\Delta n}$ inducing a single edge, where $\Delta$ is the maximum degree. This last result partially answers a question of Nakamoto and Ozeki.