Mutually orthogonal graph squares for disjoint union of paths

Abstract

This paper gives some new results on mutually orthogonal graph squares (MOGS). These generalize mutually orthogonal Latin squares in an interesting way. As such, the topic is quite nice and should have broad appeal. MOGS have strong connections to core fields of finite algebra, cryptography, finite geometry, and design of experiments. We are concerned with the mutually orthogonal half starters method to construct the mutually orthogonal graph squares for disjoint union of paths.

Generalization of MacNeish’s Kronecker product theorem of mutually orthogonal Latin squares

“…El-Shanawany et al [9] generalized the Kronecker product of Latin squares as follows, if Nðm, GÞ ! k 1 and Nðn, HÞ !…”

“…Let G be a subgraph of K n, n with n edges. A square matrix L of order n is a G-square if every element in Z n ¼ f0, 1, :::, n À 1g occurs exactly n times and the graphs G j where j 2 Z n with EðG j Þ ¼ fðx, yÞ : Lðx, yÞ ¼ jg are isomorphic to G. It is clear that the G-square represents the edge decomposition of K n, n by G. The rows of an n  n square are labeled with the elements of Z n  f0g, and the columns are labeled with the elements of Z n  f1g: ðiiÞ Three mutually orthogonal 2K 1, 2 -squares, N 0 , N 1 , and N 2 of order 4, see [9]. For more illustration, see Figure 1.…”

“…3 has been computed, where G represents disjoint copies of certain subgraphs of K n, n : Higazy [11] determined N(n, G) for all possible subgraphs G with n-edges when n ¼ 3, 4. El-Shanawany and El-Mesady [9] defined the Kronecker product of MOGS of the complete bipartite graphs and used this method to construct the new mutually orthogonal disjoint union of some complete bipartite graphs squares. The authors of [10] have constructed the MOGS for disjoint union of paths.…”

“…El-Shanawany and El-Mesady [9] defined the Kronecker product of MOGS of the complete bipartite graphs and used this method to construct the new mutually orthogonal disjoint union of some complete bipartite graphs squares. The authors of [10] have constructed the MOGS for disjoint union of paths. Mutually orthogonal certain graph squares have been built in [9].…”

“…The authors of [10] have constructed the MOGS for disjoint union of paths. Mutually orthogonal certain graph squares have been built in [9]. There are close connections between the MOGS and the orthogonal arrays [12].…”