Abstract:

In 1937 MacLane gave the important theory on cycle basis: gaph G is planar if and only if G  has a 2-basis. The join G = G_1\vee G_2 of graphs G_1 and G_2 is obtained from  G_1\bigcup G_2 by adding all the edges in {(u,v)|u\in V(G_1), v\in V(G_2)}. In this paper we investigate the  basis number of G = G_1\vee G_2 and obtain an upper bound which improves the bound given by Zare. Based on this, a better bound of C_m \vee C_n is derived too.

Key words: join of graph, cycle space, basis