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October 2007 The Basis Number of the Strong Product of Paths and Cycles with Bipartite Graphs
M. M. M. Jaradat
Missouri J. Math. Sci. 19(3): 219-230 (October 2007). DOI: 10.35834/mjms/1316032980

Abstract

The basis number of a graph $G$ is defined to be the least integer $d$ such that there is a basis $\mathcal{B}$ of the cycle space of $G$ such that each edge of $G$ is contained in at most $d$ members of $\mathcal{B}$. MacLane [13] proved that a graph $G$ is planar if and only if the basis number of $G$ is less than or equal to 2. Ali [3] proved that the basis number of the strong product of a path and a star is less than or equal to 4. In this work, (1) We give an appropriate decomposition of trees. (2) We give an upper bound of the basis number of a cycle and a bipartite graph. (3) We give an upper bound of the basis number of a path and a bipartite graph. This is a generalization of Ali's result [3].

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M. M. M. Jaradat. "The Basis Number of the Strong Product of Paths and Cycles with Bipartite Graphs." Missouri J. Math. Sci. 19 (3) 219 - 230, October 2007. https://doi.org/10.35834/mjms/1316032980

Information

Published: October 2007
First available in Project Euclid: 14 September 2011

zbMATH: 1146.05031
MathSciNet: MR2348651
Digital Object Identifier: 10.35834/mjms/1316032980

Subjects:
Primary: 05C38
Secondary: 05C75

Rights: Copyright © 2007 Central Missouri State University, Department of Mathematics and Computer Science

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Vol.19 • No. 3 • October 2007
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