DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

Discussiones Mathematicae Graph Theory

PDF

Discussiones Mathematicae Graph Theory 25(3) (2005) 391-406
DOI: 10.7151/dmgt.1291

AN UPPER BOUND OF THE BASIS NUMBER OF THE STRONG PRODUCT OF GRAPHS

Mohammed M.M. Jaradat

Department of Mathematics
Yarmouk University
Irbid-Jordan
e-mail: mmjst4@yu.edu.jo

Abstract

The basis number of a graph G is defined to be the least integer d such that there is a basis B of the cycle space of G such that each edge of G is contained in at most d members of B. In this paper we give an upper bound of the basis number of the strong product of a graph with a bipartite graph and we show that this upper bound is the best possible.

Keywords: basis number; cycle space; strong product.

2000 Mathematics Subject Classification: 05C38, 05C75.

References

[1] A.A. Ali, The basis number of complete multipartite graphs, Ars Combin. 28 (1989) 41-49.
[2] A.A. Ali and G.T. Marougi, The basis number of the strong product of graphs, Mu'tah Lil-Buhooth Wa Al-Dirasat 7 (1) (1992) 211-222.
[3] A.A. Ali and G.T. Marougi, The basis number of cartesian product of some graphs, J. Indian Math. Soc. 58 (1992) 123-134.
[4] A.S. Alsardary, An upper bound on the basis number of the powers of the complete graphs, Czechoslovak Math. J. 51 (126) (2001) 231-238, doi: 10.1023/A:1013734628017.
[5] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (America Elsevier Publishing Co. Inc., New York, 1976).
[6] R. Diestel, Graph Theory, Graduate Texts in Mathematics, 173 (Springer-Verlag, New York, 1997).
[7] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (Wiley, New York, 2000).
[8] W. Imrich and P. Stadler, Minimum cycle bases of product graphs, Australas. J. Combin. 26 (2002) 233-244.
[9] M.M.M. Jaradat, On the basis number of the direct product of graphs, Australas. J. Combin. 27 (2003) 293-306.
[10] M.M.M. Jaradat, The basis number of the direct product of a theta graph and a path, Ars Combin. 75 (2005) 105-111.
[11] P.K. Jha, Hamiltonian decompositions of product of cycles, Indian J. Pure Appl. Math. 23 (1992) 723-729.
[12] P.K. Jha and G. Slutzki, A note on outerplanarity of product graphs, Zastos. Mat. 21 (1993) 537-544.
[13] S. MacLane, A combinatorial condition for planar graphs, Fundamenta Math. 28 (1937) 22-32.
[14] G. Sabidussi, Graph multiplication, Math. Z. 72 (1960) 446-457, doi: 10.1007/BF01162967.
[15] E.F. Schmeichel, The basis number of a graph, J. Combin. Theory (B) 30 (1981) 123-129, doi: 10.1016/0095-8956(81)90057-5.
[16] P.M. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc. 13 (1962) 47-52, doi: 10.1090/S0002-9939-1962-0133816-6.

Received 29 June 2004
Revised 3 January 2005