Thrackles on nonplanar surfaces

TL;DR

This paper disproves a conjecture extending thrackle bounds from the plane to higher-genus surfaces, showing that graphs can have significantly more edges when thrackled on such surfaces than previously conjectured.

Contribution

The paper demonstrates that the existing conjecture on thrackle bounds for orientable surfaces is false, providing explicit constructions and bounds for thrackles on higher-genus surfaces.

Findings

Existence of graphs with 2n+2g-8 edges thrackled on S_g

Disproof of Cairns and Nikolayevsky's conjecture for g>0

Bounds on thrackleability of complete bipartite and complete graphs

Abstract

A thrackle is a drawing of a graph on a surface such that (i) adjacent edges only intersect at their common vertex; and (ii) nonadjacent edges intersect at exactly one point, at which they cross. Conway conjectured that if a graph with $n$ vertices and $m$ edges can be thrackled on the plane, then $m\le n$. Conway's conjecture remains open; the best bound known is that $m\le 1.393n$. Cairns and Nikolayevsky extended this conjecture to the orientable surface $S_g$ of genus $g > 0$, claiming that if a graph with $n$ vertices and $m$ edges has a thrackle on $S_g$, then $m \le n + 2g$. We disprove this conjecture. In stark contrast with the planar case, we show that for each $g>0$ there is a connected graph with $n$ vertices and $2n + 2g -8$ edges that can be thrackled on $S_g$. This leaves relatively little room for further progress involving thrackles on orientable surfaces, as every connected graph with $n$ vertices and $m$ edges that can be thrackled on $S_g$ satisfies that $m \le 2n + 4g - 2$. We prove a similar result for nonorientable surfaces. We also derive nontrivial upper and lower bounds on the minimum $g$ such that $K_{m,n}$ and $K_n$ can be thrackled on $S_g$.