This is the first of a series of several papers dealing with various combinatorial properties of triangulations. The problems generally have their origin in combinatorial theory, but are illuminated by viewing them from a more topological viewpoint.
Successive papers will deal with a generalization of four colorings to nonsimply connected surfaces (using discrete fiber bundles); analogs of coloring based on regular polyhedra other than the tetrahedron, triangulations of surfaces whose vertex degrees are all divisible by a fixed integer, homotopy and cobordism problems of the preceeding structures, and generalizations to dimensions greater than 2.
1. DEFINITIONS
A four coloring of a triangulation M is a simplicial mapf: M -+ &I3 with the property that the image of any triangle of M is a triangle of ZU13. Such a map we call a nondegenerate simplicial map. By &13 we mean the boundary of the tetrahedron, so it has exactly four vertices. We consider two four colorings f and g to be the same if they differ by a permutation of vertices. That is, there is an automorphism o of &I3 such that uf = g.
The two-manifold M is a sphere, except in Section 5, where it may be any orientable two-manifold. We assume an orientation for M has been fixed.
If M is a triangulation, and p a vertex of M, the degree of p, written p(p), is the number of triangles of M containingp. A vertex is odd (even) in M iff its degree is odd (even). A three coloring of M is a simplicial map f: M -+ A2 such that every triangle maps to a triangle. (Here LIP is the 326 Copyright