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 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.

Grünbaum coloring and its generalization to arbitrary dimension

“…It is a conjecture of Mohar [38] that the answer to this question is yes. As shown in [34], a combination of known results by Ringel [40], Youngs [54], Grannell, Griggs, and Širáň [21], Grannell and Korzhik [23] on facet (2-face) 2-colorability along with Theorem 1 leads to the following corollary.…”

“…The proof is an application of a classical theorem of König [33,30,37] in the right place at the right time. As observed in [34], König's Theorem entails that each bipartite ℓ-regular graph is edge ℓ-colorable. Since a graph is bipartite if and only if it is vertex 2-colorable, it follows that if T is facet 2-colorable, then A(T ) is an (n + 1)-regular bipartite graph and by König's Theorem admits an edge (n + 1)-coloring which naturally induces a Grünbaum hyper-coloring of T .…”

“…The notion of Grünbaum coloring is generalized in [34] for an arbitrary face size ℓ 3. In this paper we consider only triangulations (ℓ = 3 in the case of dimension 2 BASIC OBSERVATION n = 2) but generalize in another direction: we extend the notion of Grünbaum coloring from the 2-dimensional case to the general case of arbitrary (finite) dimension n 1.…”

“…The 2-dimensional version (n = 2) of Theorem 1 was established in [34] in a more general setting of ℓ-angulations.…”

“…A Tait coloring is a usual edge 3-coloring of a 3-regular graph. By Vizing's Theorem [52], χ ′ (G * (T )) ∈ {3, 4} and so four colors will certainly suffice for coloring the edges of any triangulation T of any closed 2-manifold so that the three edges have distinct colors around each face of T , which is mentioned in [30,34]. There are other types of cyclic colorations; for instance, one may color the faces of T so that the face colors are all distinct around each vertex of T ; see [12].…”