PDF
Discussiones Mathematicae Graph Theory 26(1) (2006)
113-134
DOI: https://doi.org/10.7151/dmgt.1306
ON THE BASIS NUMBER AND THE MINIMUM CYCLE BASES OF THE WREATH PRODUCT OF SOME GRAPHS I
Mohammed M.M. Jaradat
Department of Mathematics
Yarmouk University
Irbid-Jordan
e-mail: mmjst4@yu.edu.jo
Abstract
A construction of a minimum cycle bases for the wreath product of some classes of graphs is presented. Moreover, the basis numbers for the wreath product of the same classes are determined.Keywords: cycle space, basis number, cycle basis, wreath product.
2000 Mathematics Subject Classification: 05C38, 05C75.
References
[1] | M. Anderson and M. Lipman, The wreath product of graphs, in: Graphs and Applications (Boulder, Colo., 1982), (Wiley-Intersci. Publ., Wiley, New York, 1985) 23-39. |
[2] | A.A. Ali, The basis number of complete multipartite graphs, Ars Combin. 28 (1989) 41-49. |
[3] | A.A. Ali, The basis number of the direct product of paths and cycles, Ars Combin. 27 (1989) 155-163. |
[4] | A.A. Ali and G.T. Marougi, The basis number of cartesian product of some graphs, J. Indian Math. Soc. 58 (1992) 123-134. |
[5] | A.S. Alsardary and J. Wojciechowski, The basis number of the powers of the complete graph, Discrete Math. 188 (1998) 13-25, doi: 10.1016/S0012-365X(97)00271-9. |
[6] | J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (America Elsevier Publishing Co. Inc., New York, 1976). |
[7] | W.-K. Chen, On vector spaces associated with a graph, SIAM J. Appl. Math. 20 (1971) 525-529, doi: 10.1137/0120054. |
[8] | D.M. Chickering, D. Geiger and D. HecKerman, On finding a cycle basis with a shortest maximal cycle, Inform. Process. Lett. 54 (1994) 55-58, doi: 10.1016/0020-0190(94)00231-M. |
[9] | L.O. Chua and L. Chen, On optimally sparse cycles and coboundary basis for a linear graph, IEEE Trans. Circuit Theory 20 (1973) 54-76. |
[10] | G.M. Downs, V.J. Gillet, J.D. Holliday and M.F. Lynch, Review of ring perception algorithms for chemical graphs, J. Chem. Inf. Comput. Sci. 29 (1989) 172-187, doi: 10.1021/ci00063a007. |
[11] | W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (Wiley, New York, 2000). |
[12] | W. Imrich and P. Stadler, Minimum cycle bases of product graphs, Australas. J. Combin. 26 (2002) 233-244. |
[13] | M.M.M. Jaradat, On the basis number of the direct product of graphs, Australas. J. Combin. 27 (2003) 293-306. |
[14] | M.M.M. Jaradat, The basis number of the direct product of a theta graph and a path, Ars Combin. 75 (2005) 105-111. |
[15] | M.M.M. Jaradat, An upper bound of the basis number of the strong product of graphs, Discuss. Math. Graph Theory 25 (2005) 391-406, doi: 10.7151/dmgt.1291. |
[16] | M.M.M. Jaradat, M.Y. Alzoubi and E.A. Rawashdeh, The basis number of the Lexicographic product of different ladders, SUT J. Math. 40 (2004) 91-101. |
[17] | A. Kaveh, Structural Mechanics, Graph and Matrix Methods. Research Studies Press (Exeter, UK, 1992). |
[18] | G. Liu, On connectivities of tree graphs, J. Graph Theory 12 (1988) 435-459, doi: 10.1002/jgt.3190120318. |
[19] | S. MacLane, A combinatorial condition for planar graphs, Fundamenta Math. 28 (1937) 22-32. |
[20] | M. Plotkin, Mathematical basis of ring-finding algorithms in CIDS, J. Chem. Doc. 11 (1971) 60-63, doi: 10.1021/c160040a013. |
[21] | 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. |
[22] | P. Vismara, Union of all the minimum cycle bases of a graph, Electr. J. Combin. 4 (1997) 73-87. |
[23] | D.J.A. Welsh, Kruskal's theorem for matroids, Proc. Cambridge Phil, Soc. 64 (1968) 3-4, doi: 10.1017/S030500410004247X. |
Received 3 March 2005
Revised 9 September 2005
Close