Thrackles on nonplanar surfaces

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 $S_g$.