联图的圈基

The basis number of join graphs

MacLane于1937年给出了圈基方面的重要定理:图G是平面图,当且仅当图G有2-重基.连通图G_1和G_2的联图G_1∨G_2指的是在它们的不交并G_1∪G_2上添加边集{(u,v)|u∈V(G_1),v∈V(G_2)}对G_1和G_2的联图G_1∨G_2的圈基重数进行了研究,得到了一个上界,改进了Zare的结果.并在此基础之上,进一步得到特殊联图C_m∨C_n的圈基重数的一个上界.

In 1937 MacLane gave the important theory on cycle basis: Graph G is planar if and only if G has a 2-basis. The join G=G1 ∨ G2 of graphs G1 and G2 is obtained from G1∪G2 by adding all the edges in {(u,v)|u∈V(G1), v ∈ V(G2)}. In this paper we investigate the basis number of G=G1∨G2 and obtain an upper bound which improves the bound given by Zare. Based on this,a better bound of Cm ∨ Cn is derived too.