Thep-intersection number of a complete bipartite graph and orthogonal double coverings of a clique

Abstract

Thep-intersection graph of a collection of finite sets {S _i } ⁿ _i=1 is the graph with vertices 1, ...,n such thati, j are adjacent if and only if |S _i ∩S _j |≥p. Thep-intersection number of a graphG, herein denoted θ _p (G), is the minimum size of a setU such thatG is thep-intersection graph of subsets ofU. IfG is the complete bipartite graphK _n,n andp≥2, then θ _p (K _{n, n} )≥(n ²+(2p−1)n)/p. Whenp=2, equality holds if and only ifK _n has anorthogonal double covering, which is a collection ofn subgraphs ofK _n , each withn−1 edges and maximum degree 2, such that each pair of subgraphs shares exactly one edge. By construction,K _n has a simple explicit orthogonal double covering whenn is congruent modulo 12 to one of {1, 2, 5, 7, 10, 11}.