Genus distribution polynomials for bicellular bicolored maps all with real zeros

TL;DR

This paper derives explicit formulas for genus distribution polynomials of bicellular bicolored maps, demonstrating they have only real zeros, which implies the genus distribution is log-concave, advancing combinatorial map enumeration.

Contribution

It provides the first explicit formula for genus distribution polynomials of bicellular bicolored maps with arbitrary face length, showing they have only real zeros.

Findings

Genus distribution polynomials have only real zeros.

Genus distribution is log-concave.

Explicit formulas for bicellular bicolored maps are obtained.

Abstract

Enumerating bicolored maps and maps according to the numbers (and possibly types) of edges, faces, white vertices, black vertices and genus has been an important topic arising in many fields of mathematics and physics. In particular, Jackson (1987), Zagier (1995) and Stanley (2011) respectively obtained some expressions for the generating polynomial of the numbers of one-face bicolored maps with given number of edges and white vertex degree distribution while tracking the number of black vertices. The cases for multiple faces are harder. In this paper, we first obtain the number for that of bicolored maps with two faces, i.e., bicellular, of arbitrary length distribution, and then derive an explicit formula for the corresponding generating polynomial with respect to genus. We next prove that the generating polynomial essentially has only real zeros and thus the genus distribution is log-concave.