Planar Eulerian triangulations are equivalent to spherical Latin bitrades.

Journal of Combinatorial Theory Ser. A 115 (2008), 193-197.

Given a pair of latin squares, we may remove from both squares those cells that contain the same symbol in corresponding positions. The resulting pair $T=\{P_1,P_2\}$ of partial latin squares is called a latin bitrade. The number of filled cells in $P_1$ is called the size of $T$. There are at least two natural ways to define the genus of a latin bitrade; the bitrades of genus 0 are called spherical. We construct a simple bijection between the isomorphism classes of planar Eulerian triangulations on $v$ vertices and the main classes of spherical latin bitrades of size $v-2$. Since there exists a fast algorithm (due to Batagelj, Brinkmann and McKay) for generating planar Eulerian triangulations up to isomorphism, our result implies that also spherical latin bitrades can be generated very efficiently.