Unsensed enumeration of cubic unicellular maps on orientable and non-orientable surfaces
Alexander Omelchenko, Igor Labutin
TL;DR
The paper addresses the problem of unsensed enumeration of cubic unicellular maps on orientable and non-orientable closed surfaces by extending the orbifold framework to this specialized regular class. The main approach reduces unsensed counts to quotient maps and to rooted precubic counts, yielding explicit closed formulas such as $\bar{\tau}_+^{(3)}(g)$ for orientable hosts and a completely explicit finite-sum expression for $\bar{\tau}^{(3)}_-(g)$ in the non-orientable case, with all coefficients computable and rooted precubic inputs explicit. Numerical data up to genus $g=20$ validate the formulas, and asymptotics show precise leading-factor behavior: for orientable hosts, $\widetilde{\tau}^{(3)}_+(g) \sim \tfrac{1}{2E}\,\tau^{(3)}_+(g)$ and $\bar{\tau}^{(3)}_+(g) \sim \tfrac{1}{4E}\,\tau^{(3)}_+(g)$ with $E=6g-3$, while for non-orientable hosts $\bar{\tau}^{(3)}_-(g) \sim \tfrac{1}{4(3g-3)}\,\tau_-^{(3)}(g)$. The work provides complete explicit unsensed enumeration for cubic unicellular maps on both surface types and supplies extensive numerical data supporting the theory.
Abstract
We enumerate cubic (3-regular) unicellular maps on closed surfaces up to all homeomorphisms. Using the orbifold approach, we reduce the unsensed enumeration to explicit counts of quotient maps and rooted cubic/precubic maps on simpler surfaces. For orientable hosts this yields a compact identity expressed through known sensed and rooted numbers; for non orientable hosts we obtain a fully explicit finite sum expression via precubic counts. Numerical tables are provided, together with a brief asymptotic discussion.