Higher-dimensional generalization of Youngs' theorem and circular colorings

TL;DR

This paper generalizes Youngs' theorem on quadrangulations and 3-chromaticity to higher dimensions and broader classes of surfaces, extending its applicability to non-orientable surfaces and circular colorings.

Contribution

It introduces a higher-dimensional generalization of Youngs' theorem, unifying and extending previous results on graph colorings and surface embeddings.

Findings

Generalization of Youngs' theorem to higher dimensions

Extension to non-orientable surfaces and circular colorings

Unified framework for various surface coloring results

Abstract

In 1996, Youngs proved that any quadrangulation of the real projective plane is not 3-chromatic. This result has been extended in various directions over the years, including to other non-orientable closed surfaces, higher-dimensional analogues of quadrangulations and circular colorings. In this paper, we provide a generalization which yields some of these extensions of Youngs' theorem.