All face 2-colorable d-angulations are Gr\"unbaum colorable

TL;DR

The paper generalizes the concept of Gr"unbaum coloring from triangulations to all face sizes in $d$-angulations, proving that face 2-colorability implies Gr"unbaum colorability.

Contribution

It introduces a generalization of Gr"unbaum coloring for arbitrary $d$-angulations and establishes that face 2-colorability ensures Gr"unbaum colorability.

Findings

Face 2-colorability implies Gr"unbaum colorability for $d$-angulations.

Certain classes of triangulations are shown to be face 2-colorable.

The concept of Gr"unbaum coloring is extended beyond triangulations.

Abstract

A $d$-angulation of a surface is an embedding of a 3-connected graph on that surface that divides it into $d$-gonal faces. A $d$-angulation is said to be Gr\"unbaum colorable if its edges can be $d$-colored so that every face uses all $d$ colors. Up to now, the concept of Gr\"unbaum coloring has been related only to triangulations ($d = 3$), but in this note, this concept is generalized for an arbitrary face size $d \geqslant 3$. It is shown that the face 2-colorability of a $d$-angulation $P$ implies the Gr\"unbaum colorability of $P$. Some wide classes of triangulations have turned out to be face 2-colorable.