A Note on $4$-colorings of Quadrangulations

TL;DR

This paper proves that in any quadrangulation on an orientable surface with a proper 4-coloring, the number of faces with a specific boundary color order equals the number with the reverse order, implying an even count of rainbow-faces.

Contribution

It establishes a symmetry property of rainbow faces in quadrangulations, revealing that the count of faces with reversed boundary color sequences are equal.

Findings

Number of (c1,c2,c3,c4)-faces equals the number of (c4,c3,c2,c1)-faces.

Number of rainbow-faces in G is even.

Symmetry property holds for faces with specific boundary color sequences.

Abstract

Let $G$ be a quadrangulation on an orientable surface and let $g$ be a proper vertex-$4$-coloring of $G$. A face $F$ of $G$ is said to be a rainbow-face if all four distinct colors appear on its boundary. A $(c_1,c_2,c_3,c_4)$-face in $G$ is a rainbow face with colors $c_i$, $i=1,2,3,4$ on the boundary in clockwise order. We show that the number of $(c_1,c_2,c_3,c_4)$-faces in $G$ equals the number of $(c_4,c_3,c_2,c_1)$-faces. This implies in particular that the number of rainbow-faces of $G$ is even.