Mutually orthogonal graph squares

Abstract

AbstractA decomposition 𝒢={G1, G2,…,Gs} of a graph G is a partition of the edge set of G into edge‐disjoint subgraphs G1, G2,…,Gs. If Gi≅H for all i∈{1, 2, …, s}, then 𝒢 is a decomposition of G by H. Two decompositions 𝒢={G1, G2, …, Gn} and ℱ={F1, F2,…,Fn} of the complete bipartite graph Kn,n are orthogonal if |E(Gi)∩E(Fj)|=1 for all i,j∈{1, 2, …, n}. A set of decompositions {𝒢1, 𝒢2, …, 𝒢k} of Kn, n is a set of k mutually orthogonal graph squares (MOGS) if 𝒢i and 𝒢j are orthogonal for all i, j∈{1, 2, …, k} and i≠j. For any bipartite graph G with n edges, N(n, G) denotes the maximum number k in a largest possible set {𝒢1, 𝒢2, …, 𝒢k} of MOGS of Kn, n by G. El‐Shanawany conjectured that if p is a prime number, then N(p, Pp+ 1)=p, where Pp+ 1 is the path on p+ 1 vertices. In this article, we prove this conjecture. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 369–373, 2009