On Mutually Orthogonal Graph-Path Squares


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).

Share and Cite:

El-Shanawany, R. (2016) On Mutually Orthogonal Graph-Path Squares. Open Journal of Discrete Mathematics, 6, 7-12. doi: 10.4236/ojdm.2016.61002.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Balakrishnan, R. and Ranganathan, K. (2012) A Textbook of Graph Theory. Springer, Berlin.
[2] Alspach, B., Heinrich, K. and Liu, G. (1992) Orthogonal Factorizations of Graphs. In: Dinitz, J.H. and Stinson, D.R., Eds., Contemporary Design Theory, Chapter 2, Wiley, New York, 13-40.
[3] Gronau, H.-D.O.F., Hartmann, S., Grüttmüller, M., Leck, U. and Leck, V. (2002) On Orthogonal Double Covers of Graphs. Designs, Codes and Cryptography, 27, 49-91.
[4] Colbourn, C.J. and Dinitz, J.H. (eds.) (2007) Handbook of Combinatorial Designs. 2nd Edition, Chapman & Hall/CRC, London, Boca Raton.
[5] Colbourn, C.J. and Dinitz, J.H. (2001) Mutually Orthogonal Latin Squares: A Brief Survey of Constructions. Journal of Statistical Planning and Inference, 95, 9-48.
[6] El-Shanawany, R. (2002) Orthogonal Double Covers of Complete Bipartite Graphs. Ph.D. Thesis, Universitat Rostock, Rostock.
[7] Sampathkumar, R. and Srinivasan, S. (2009) Mutually Orthogonal Graph Squares. Journal of Combinatorial Designs, 17, 369-373.
[8] El-Shanawany, R., Gronau, H.-D.O.F. and Grüttmüller, M. (2004) Orthogonal Double Covers of Kn,n by Small Graphs. Discrete Applied Mathematics, 138, 47-63.
[9] El-Shanawany, R., Shabana, H. and ElMesady, A. (2014) On Orthogonal Double Covers of Graphs by Graph-Path and Graph-Cycle. LAP LAMBERT Academic Publishing.

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.