An upper bound of the basis number of the semi-strong product of bipartite graphs

Abstract

A basis of the cycle space, $\mathcal{C}(G)$, of a graph $G$ is called a $d$-fold if each edge of $G$ occurs in at most $d$ cycles of the basis. The basis number, $b\left(G\right)$, of a graph $G$ is defined to be the least integer $d$ such that $G$ has a $d$-fold basis for its cycle space. MacLane proved that a graph $G$ is planar if and only if $b\left(G\right)\le 2$. Schmeichel showed that for $n\ge 5$, $b(K_n•P_2)\le 1+b(K_n)$ Ali proved that for $n,m\ge 5$, $b(K_n•K_m)\le 3+b(K_m)+b(K_n)$. Jaradat proved that for any two bipartite graphs $G$ and $H$, $b(G\text{Λ}H)\le 5+b(G)+b(H)$. In this paper we give an upper bound of the basis number of the semi-strong product of bipartite graphs. Also, we give an example where the bound is achieved.