The intersection number of complete r-partite graphs

Abstract

Latin squares C, D of order n are called pseudo-orthogonal if any two rows of the matrices C and D have exactly one common element. We give conditions for existence of families consisting of t pseudo-orthogonal Latin squares of order n. It is proved that the intersection number of a complete r-partite graph equals n2 if and only if there exists a family consisting of r – 2 pairwise pseudo-orthogonal Latin squares of order n. It is proved that if , where prols(n) is the maximum t such that there exists a set of t pseudo-orthogonal Latin squares of order n, then the intersection number of the graph is equal to n2. Applications of the obtained results to calculating the intersection number of some graphs are given.