On high genus extensions of Negami's conjecture

TL;DR

This paper extends Negami's conjecture to higher genus surfaces, establishing conditions for graph embeddability in such surfaces and proving related decidability and boundedness results.

Contribution

It introduces a natural extension of Negami's conjecture to all compact surfaces, including orientable and non-orientable, and proves boundedness and decidability results for these embeddings.

Findings

Graphs with finite covers embeddable in a surface have bounded Euler genus.

Decidability of the extended conjecture for surfaces with large Euler genus.

Equivalence of embeddability in a surface and existence of a finite ply cover for countable graphs.

Abstract

Negami's famous planar cover conjecture is equivalent to the statement that a connected graph can be embedded in the projective plane if and only if it has a projective planar cover. In 1999, Hlin\v{e}n\'y proposed extending this conjecture to higher genus non-orientable surfaces. In this paper, we put forward a natural extension that encompasses orientable surfaces as well; for every compact surface $\Sigma$, a connected graph $G$ has a finite cover embeddable in $\Sigma$ if and only if $G$ is embeddable in a surface covered by $\Sigma$. As evidence toward this, we prove that for every surface $\Sigma$, the connected graphs with a finite cover embeddable in $\Sigma$ have bounded Euler genus. Moreover, we show that these extensions of Negami's conjecture are decidable for every compact surface of sufficiently large Euler genus, surpassing what is known for Negami's original conjecture. We also prove the natural analogue for countable graphs embeddable into a compact (orientable) surface. More precisely, we prove that a connected countable graph $G$ has a finite ply cover that embeds into a compact (orientable) surface if and only if $G$ embeds into a compact (orientable) surface. Our most general theorem, from which these results are derived, is that there is a constant $c>0$ such that for every surface $\Sigma$, there exists a decreasing function $p_\Sigma:\mathbb{N} \to \mathbb{N}$ with $\lim_{g\to \infty}p_\Sigma(g) =0$ such that every finite cover embeddable in $\Sigma$ of any connected graph with Euler genus $g\ge c$ has ply at most $p_\Sigma(g)$.