The Basis Number of the Strong Product of Paths and Cycles with Bipartite Graphs

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].