Generalized Hadamard Matrices and 2-Factorization of Complete Graphs

Graph factorization plays a major role in graph theory and it shares common ideas in important problems such as edge coloring and Hamiltonian cycles. A factor of a graph is a spanning subgraph of which is not totally disconnected. An - factor is an - regular spanning subgraph of and is -factorable if there are edge-disjoint -factors such that . We shall refer as an -factorization of a graph . In this research we consider -factorization of complete graph. A graph with vertices is called a complete graph if every pair of distinct vertices is joined by an edge and it is denoted by . We look into the possibility of factorizing with added limitations coming in relation to the rows of generalized Hadamard matrix over a cyclic group. Over a cyclic group of prime order , a square matrix of order all of whose elements are the root of unity is called a generalized Hadamard matrix if , where is the conjugate transpose of matrix and is the identity matrix of order . In the present work, generalized Hadamard matrices over a cyclic group have been considered. We prove that the factorization is possible for in the case of the limitation 1, namely, If an edge belongs to the factor , then the and entries of the corresponding generalized Hadamard matrix should be different in the row. In Particular, number of rows in the generalized Hadamard matrices is used to form -factorization of complete graphs. We discuss some illustrative examples that might be used for studying the factorization of complete graphs.

On Symmetric Transversal Designs STD8[24; 3]'s

The existence of a class regular symmetric transversal design STDλ[3λ;3] is equivalent to a generalized Hadamard matrix of order 3u over GF(3). Let nλ be the number of nonisomorphic STDλ[3λ;3]'s. It is known that n1=1, n2=1, n3=4, n4=1, n5=0, n6≥20, and n7≥5. In this paper, it is shown that n8≥24.