## Local colorings of triangulations of orientable surfaces

Let G be a triangulation of some surface. For a vertex *v* of G, let S(*v*)
(the *star* of *v*) denote the set of all facial triangles containing *v*.
A *local coloring* is a collection of 4-colorings of all stars of vertices
of G such that for every edge *uv* of G, there is a permutation of colors
such that the coloring of S(*v*) and the permuted coloring of S(*u*) agree
on two triangles they have in common. This notion of a coloring is equivalent to the usual vertex 4-coloring for
planar triangulations, but is
not equivalent for non-simply-connected surfaces.

**
****Problem (Steve Fisk):** Does every triangulation of an
orientable surface have a local
coloring?

Why the restriction to orientable surfaces? The graph K_{6}
embedded in the projective plane has no local coloring. For more information see
[1,2].

[1] S. Fisk, Combinatorial structures on triangulations. II. Local colorings,
Adv. Math. **11 **(1973), 339-350.

[2] S. Fisk, Variations on coloring, surfaces and higher-dimensional
manifolds, Adv. Math. **25 **(1977), 226-266.

Send comments to Bojan.Mohar@uni-lj.si

##### Revised: maj 13, 2002.