Abstract:Suppose that $T^{\circ }$ and $T^{\star }$ are partial latin squares of order $n$, with the property that each row and each column of $T^{\circ }$ contains the same set of entries as the corresponding row or column of $T^{\star }$. In addition, suppose that each cell in $T^{\circ }$ contains an entry if and only if the corresponding cell in $T^{\star }$ contains an entry, and these entries (if they exist) are different. Then the pair $T=(T^{\circ },T^{\star })$ forms a {latin bitrade}. The {size} of $T$ is the total number of filled cells in $T^{\circ }$ (equivalently $T^{\star }$). The latin bitrade is {minimal} if there is no latin bitrade $(U^{\circ },U^{\otimes })$ such that $U^{\circ }\subseteq T^{\circ }$. Dr\'apal (2003) represented latin bitrades in terms of row, column and entry cycles, which he proved formed a coherent digraph. This digraph can be considered as a combinatorial surface, thus associating each latin bitrade with an integer genus, which is a robust structural property of the latin bitrade. For each genus $g\ge 0$, we construct a latin bitrade of smallest possible size, and also a minimal latin bitrade of size $8g+8$.
Keywords: latin trade, bitrade, genus
AMS Subject Classification: 05B15