Open Journal of Discrete Mathematics, 2016, 6, 7-12
Published Online January 2016 in SciRes. http://www.scirp.org/journal/ojdm
http://dx.doi.org/10.4236/ojdm.2016.61002
How to cite this paper: El-Shanawany, R. (2016) On Mutually Orthogonal Graph-Path Squares. Open Journal of Discrete
Mathematics, 6, 7-12. http://dx.doi.org/10.4236/ojdm.2016.61002
On Mutually Orthogonal Graph-Path Squares
Ramadan El-Shanawany
Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University,
Menouf, Egypt
Received 14 January 2015; accepted 14 December 2015; published 17 December 2015
Copyright © 2016 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
A decomposition { }
s
G G G
−
=
0 1 1
, , , of a graph H is a partition of the edge set of H into edge-
disjoint subgraphs
s
G G G
− 0 1 1
, , , . If
i
G G ≅ for all { } i s ∈ − 0,1, , 1 , then is a decomposition
of H by G. Two decompositions { }
n
G G G
−
=
0 1 1
, , , and { }
n
F F F
−
=
0 1 1
, , , of the complete
bipartite graph
nn
K
,
are orthogonal if, ( ) ( )
i j
EG F = 1 for all { } ij n ∈ − , 0,1, , 1 . A set of
decompositions { }
k − 0 1 1
, , , of
nn
K
,
is a set of k mutually orthogonal graph squares (MOGS)
if
i
and
j
are orthogonal for all { } ij k ∈ − , 0,1, , 1 and i j ≠ . For any bipartite graph G with
n edges, ( ) NnG , denotes the maximum number k in a largest possible set { }
k − 0 1 1
, , , of
MOGS of
nn
K
,
by . Our objective in this paper is to compute ( ) NnG , where ( )
d
G F
+
=
1
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).
Keywords
Orthogonal Graph Squares, Orthogonal Double Cover
1. Introduction
In this paper we make use of the usual notation:
, mn
K for the complete bipartite graph with partition sets of
sizes m and n,
1 n
P
+
for the path on n + 1 vertices, D F for the disjoint union of D and F,
v
L
D F for the
union of D and F with
v
L (set of vertices) that belong to each other (i.e. union of D and F with common ver-
tices of the set
v
L belong to F and D),
n
K for the complete graph on n vertices,
1
K for an isolated vertex.
The other terminologies not defined here can be found in [1].
A decomposition { }
0 1 1
, , ,
s
G G G
−
= of a graph H is a partition of the edge set of H into edge-disjoint sub-