Counting Colored Random Triangulations

TL;DR

This paper provides a new combinatorial derivation for counting vertex-tricolored planar triangulations, extending to k-gonal tessellations, and verifies results using matrix models.

Contribution

It introduces a direct combinatorial approach using decorated trees for counting colored triangulations, generalizing previous results and connecting to matrix models.

Findings

Derived a combinatorial counting function for tricolored triangulations

Extended enumeration methods to k-gonal tessellations with cyclic colorings

Validated results through matrix model techniques

Abstract

We revisit the problem of enumeration of vertex-tricolored planar random triangulations solved in [Nucl. Phys. B 516 [FS] (1998) 543-587] in the light of recent combinatorial developments relating classical planar graph counting problems to the enumeration of decorated trees. We give a direct combinatorial derivation of the associated counting function, involving tricolored trees. This is generalized to arbitrary k-gonal tessellations with cyclic colorings and checked by use of matrix models.