Edge-clique covers of the tensor product

In this paper we study the edge-clique cover number, θe(⋅)θe(⋅), of the tensor product Kn×KnKn×Kn. We derive an easy lowerbound for the edge-clique number of graphs in general. We prove that, when n   is prime θe(Kn×Kn)θe(Kn×Kn) matches the lowerbound. Moreover, we prove that θe(Kn×Kn)θe(Kn×Kn) matches the lowerbound if and only if a projective plane of order n   exists. We also show an easy upperbound for θe(Kn×Kn)θe(Kn×Kn) in general, and give its limiting value when the Riemann hypothesis is true. Finally, we generalize our work to study the edge-clique cover number of the higher-dimensional tensor product Kn×Kn×⋯×KnKn×Kn×⋯×Kn.