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arXiv · 2210.00671

Non-recursive Counts of Graphs on Surfaces

Nicholas Ercolani·Joceline Lega·Brandon Tippings
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Abstract

The problem of map enumeration concerns counting connected spatial graphs, with a specified number $j$ of vertices, that can be embedded in a compact surface of genus $g$ in such a way that its complement yields a cellular decomposition of the surface. As such this problem lies at the cross-roads of combinatorial studies in low dimensional topology and graph theory. The determination of explicit formulae for map counts, in terms of closed classical combinatorial functions of $g$ and $j$ as opposed to a recursive prescription, has been a long-standing problem with explicit results known only for very low values of $g$. In this paper we derive closed-form expressions for counts of maps with an arbitrary number of even-valent vertices, embedded in surfaces of arbitrary genus. In particular, we exhibit a number of higher genus examples for 4-valent maps that have not appeared prior in the literature.

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Nicholas Ercolani, Joceline Lega, Brandon Tippings. 2022-10-03. Non-recursive Counts of Graphs on Surfaces. https://arxiv.org/abs/2210.00671

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