Graphs for Orthogonal Arrays and Projective Planes of Even Order

Abstract

We consider orthogonal arrays of strength two and even order $q$ having $n$ columns which are equivalent to $n-2$ mutually orthogonal Latin squares of order $q$. We show that such structures induce graphs on $n$ vertices, invariant up to complementation. Previous methods worked only for single Latin squares of even order and were harder to apply. If $q$ is divisible by 4, the invariant graph is simple undirected. If $q$ is 2 modulo 4, the graph is a tournament. When $n=q+1$ is maximal, the array corresponds to an affine plane, and the vertex valencies of the graph have parity $q/2$ modulo 2. We give the graphs at all possible points and lines for 22 planes of order 16. Four of the planes, none of them of translation or dual translation type, produce nonempty graphs at some points.