On Mutually Orthogonal Graph-Path Squares

Ramadan El-Shanawany

doi:10.4236/ojdm.2016.61002

Abstract

A decomposition of a graph H is a partition of the edge set of H into edge-disjoint subgraphs . If for all , then G is a decomposition of H by G. Two decompositions and of the complete bipartite graph are orthogonal if, for all . A set of decompositions of is a set of k mutually orthogonal graph squares (MOGS) if and are orthogonal for all and . For any bipartite graph G with n edges, denotes the maximum number k in a largest possible set of MOGS of by G. Our objective in this paper is to compute where is a path of length d with d + 1 vertices (i.e. Every edge of this path is one-to-one corresponding to an isomorphic to a certain graph F).

Highlights

In this paper we make use of the usual notation: Km,n for the complete bipartite graph with partition sets of sizes m and n, Pn+1 for the path on n + 1 vertices,D F for the disjoint union of D and F,D Lv F for the union of D and F with Lv that belong to each other, Kn for the complete graph on n vertices, K1 for an isolated vertex.The other terminologies not defined here can be found in [1].A decomposition = {G0,G1, } Gs−1 of a graph H is a partition of the edge set of H into edge-disjoint sub-How to cite this paper: El-Shanawany, R. (2016) On Mutually Orthogonal Graph-Path Squares
A set of decompositions { 0, 1, k }−1 of Kn,n is a set of k mutually orthogonal graph squares (MOGS) if i and j are orthogonal for all i, j ∈{0,1, k − 1} and i ≠ j
For the union of D and F with Lv that belong to each other, Kn for the complete graph on n vertices, K1 for an isolated vertex

Summary

Introduction

A set of decompositions { 0, 1, , k }−1 of Kn,n is a set of k mutually orthogonal graph squares (MOGS) if i and j are orthogonal for all i, j ∈{0,1, , k −1} and i ≠ j . . A subgraph G of is half-starter if E (G) = n and the lengths of all edges in G are mutually different.

Results

Conclusion