Minimal and near-minimal critical sets in back-circulant latin squares

Abstract

A critical set in a latin square is a subset of its elements with the following properties: 1) No other latin square exists which also contains that subset. 2) No element may be deleted without destroying property 1. Let scs(n) denote the smallest possible cardinality of a critical set in an n × n latin square. It is conjectured that scs(n )= � n 2 /4� ,a nd that only the back-circulant latin square contains a critical set of this size. These conjectures have been proven for n ≤ 7. In this paper, we further conjecture that in a back-circulant latin square of size n> 4, the critical . . .. →