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

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.