The basis number of a graph $G$ is defined by Schmeichel to be the least integer $h$ such that $G$ has an $h$-fold basis for its cycle space. MacLane showed that a graph is planar if and only if its basis number is $\le 2$. Schmeichel proved that the basis number of the complete graph $K_n$ is at most $3$. We generalize the result of Schmeichel by showing that the basis number of the $d$-th power of $K_n$ is at most $2d+1$.
The basis number of a graph $G$ was defined by Schmeichel to be the least integer $h$ such that $G$ has an $h$-fold basis for its cycle space. He proved that for $m,n\ge 5$, the basis number $b(K_{m,n})$ of the complete bipartite graph $K_{m,n}$ is equal to 4 except for $K_{6,10}$, $K_{5,n}$ and $K_{6,n}$ with $n=5,6,7,8$. We determine the basis number of some particular non-planar graphs such as $K_{5,n}$ and $K_{6,n}$, $n=5,6,7,8$, and $r$-cages for $r=5,6,7,8$, and the Robertson graph.