More mutually orthogonal graph squares
Abstract
A decomposition (Equation) of a graph G is a partition of the edge set of G into edge-disjoint subgraphs G1, G2, ⋯, G 3. If Gi ≅ H for all i (Equation)., then G is a decomposition of G by H. Two decompositions (Equation). and (Equation).of the complete bipartite graph Kn, n are orthogonal if | E(Gi) ∩ E(Fj) |= 1 for ail i, j (Equation).. Aset {G1, G2, ⋯, Gk} of decompositions of Kn, n is a set of k mutually orthogonal graph squares(MOGS) if Gi and Gj are orthogonal for all i, j (Equation). and ≠ j. For any bipartite graph G with n edges, N(n, G) denotes the maximum number k in a largest possible set {G 1, G2, ⋯, Gk} of MOGS of K n, n by G. In this paper, we prove that if ρ is a prime number, then N(p, P4+(p-3) K2) < p-2, N(p, 2P3+{p-4) K2) < p-3, N(p, 2P4 +(p-6)/r2) > p-4 and N(p, 3P3 +(p-6) K2) < p-5, where for any positive integer I, P1 is the path on I vertices.
