On the intersection of three or four transversals of the back circulant latin square B_n

TL;DR

This paper generalizes the intersection properties of transversals in back circulant Latin squares, providing constructions and existence results for stable intersections of multiple transversals, and applying these to the existence of certain Latin trades.

Contribution

It introduces a generalized framework for intersecting transversals in back circulant Latin squares and constructs -way k-homogeneous Latin trades using stable intersection structures.

Findings

Constructed base designs for =3,4 transversals.

Established existence of -way k-homogeneous circulant Latin trades for odd orders.

Provided computational results and non-existence proofs for small cases.

Abstract

A paper by Cavenagh and Wanless diagnosed the possible intersection of any two transversals of the back circulant Latin square B_n, and used the result to completely determine the spectrum for 2-way k-homogeneous latin trades. We give a generalization of this problem for the intersection of \mu transversals of B_n and provide a construction for this problem, as well as providing base designs for the construction in the cases \mu= 3, 4 found by a computational search. This result is then applied to the problem of finding \mu-way k-homogeneous Latin trades. We generalize this problem to the intersection of \mu transversals of B_n such that the transversals intersect stably (that is, the intersection of any pair of transversals is independent of the choice of the pair) and show that these structures can be used to construct \mu-way k-homogeneous circulant latin trades of odd order. We provide a number of basic existence and non-existence results for \mu transversals of B_n that intersect stably, as well as the results of a computational search for small n. This is followed by the principal results of this paper; a construction that covers a large portion of the spectrum when n is sufficiently large, which requires certain base designs. These base designs are provided in the cases \mu= 3,4, which were found by a computational search. We use this result to find the existence of \mu-way k-homogeneous circulant latin trades of odd order, for \mu= 3, 4.