All face $2$-colorable $d$-angulations are Grünbaum 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 Grünbaum colorable if its edges can be $d$-colored so that every face uses all $d$ colors. Up to now, the concept of Grünbaum coloring has been related only to triangulations ($d=3$), but in this note, this concept is generalized for an arbitrary face size $d\ge 3$. It is shown that the face 2-colorability of a $d$-angulation $P$ implies the Grünbaum colorability of $P$. Some wide classes of triangulations have turned out to be face 2-colorable.