A planar Eulerian triangulation is a simple plane graph in which each face is a triangle and each vertex has even degree. Such objects are known to be equivalent to spherical Latin bitrades. (A Latin bitrade describes the difference between two Latin squares of the same order.) We give a classification in the near-regular case when each vertex is of degree $4$ or $6$ (which we call a near-homogeneous spherical Latin bitrade, or NHSLB). The classification demonstrates that any NHSLB is equal to two graphs...

A loop of order $n$ possesses at least $3n^2-3n+1$ associative triples. However, no loop of order $n>1$ that achieves this bound seems to be known. If the loop is involutory, then it possesses at least $3n^2-2n$ associative triples. Involutory loops with $3n^2-2n$ associative triples can be obtained by prolongation of certain maximally nonassociative quasigroups whenever $n-1$ is a prime greater than or equal to $13$ or $n-1=p^{2k}$, $p$ an odd prime. For orders $n\le 9$ the minimum number of associative triples is reported for both general and involutory...