All face \(2\)-colorable \(d\)-angulations are Grünbaum colorable

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 \( d \geq 3 \). 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.