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Let $T$ be a partial latin square and $L$ be a latin square with $T\subseteq L$. We say that $T$ is a latin trade if there exists a partial latin square $T^{\text{'}}$ with $T^{\text{'}}\cap T=\varnothing$ such that $(L\setminus T)\cup T^{\text{'}}$ is a latin square. A $k$-homogeneous latin trade is one which intersects each row, each column and each entry either $0$ or $k$ times. In this paper, we show the existence of $3$-homogeneous latin trades in abelian $2$-groups.

A latin trade is a subset of a latin square which may be replaced with a disjoint mate to obtain a new latin square. A $k$-homogeneous latin trade is one which intersects each row, each column and each entry of the latin square either $0$ or $k$ times. In this paper, we show that a construction given by Cavenagh, Donovan and Drápal for $3$-homogeneous latin trades in fact classifies every minimal $3$-homogeneous latin trade. We in turn classify all $3$-homogeneous latin trades. A corollary is that any $3$-homogeneous...

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...

The complete tripartite graph $K_{n,n,n}$ has $3n^2$ edges. For any collection of positive integers $x_1,x_2,\cdots ,x_m$ with ${\sum }_{i=1}^m{x}_i=3n^2$ and $x_i\ge 3$ for $1\le i\le m$, we exhibit an edge-disjoint decomposition of $K_{n,n,n}$ into closed trails (circuits) of lengths $x_1,x_2,\cdots ,x_m$.

Suppose that $T^{\circ }$ and $T^{☆}$ 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 $T^{☆}$. In addition, suppose that each cell in $T^{\circ }$ contains an entry if and only if the corresponding cell in $T^{☆}$ contains an entry, and these entries (if they exist) are different. Then the pair $T=(T^{\circ },T^{☆})$ forms a . The of $T$ is the total number of filled cells in $T^{\circ }$ (equivalently $T^{☆}$). The latin bitrade is if there is no latin bitrade...

We define a proper triangulation to be a dissection of an integer sided equilateral triangle into smaller, integer sided equilateral triangles such that no point is the vertex of more than three of the smaller triangles. In this paper we establish necessary and sufficient conditions for a proper triangulation of a convex region to exist. Moreover we establish precisely when at least two such equilateral triangle dissections exist. We also provide necessary and sufficient conditions for some convex...