Counting unicellular maps on non-orientable surfaces

TL;DR

This paper develops a bijective approach to count unicellular maps on non-orientable surfaces, deriving explicit formulas and asymptotics, and introduces novel involutions to handle non-orientability effects.

Contribution

It presents a new bijective method linking non-orientable unicellular maps to lower topological types, leading to explicit counting formulas and asymptotic results.

Findings

Derived recurrence relations for non-orientable unicellular maps.

Provided exact formulas for precubic cases.

Established asymptotic formulas for large edge counts.

Abstract

A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we present a bijective link between unicellular maps on a non-orientable surface and unicellular maps of a lower topological type, with distinguished vertices. From that we obtain a recurrence equation that leads to (new) explicit counting formulas for non-orientable unicellular maps of fixed topology. In particular, we give exact formulas for the precubic case (all vertices of degree 1 or 3), and asymptotic formulas for the general case, when the number of edges goes to infinity. Our strategy is inspired by recent results obtained by the second author for the orientable case, but significant novelties are introduced: in particular we construct an involution which, in some sense, "averages" the effects of non-orientability.