多重图的处处无零整数流

Nowhere-Zero Integer Flows in the Multigraphs

【目的】探讨多重图与它的基图的处处无零整数流的存在性之间的关系。【方法】用原图的处处无零整数流构造新图的处处无零整数流。【结果】1)若一个多重图的基图存在处处无零k-流,则它也存在处处无零k-流;2)如果两个多重图有共同的基图,并且它们的任意一条公共边都是基图中某个边的平行边,那么这两个多重图的处处无零k-流的存在性一致。【结论】得到的结果不但有助于对处处无零整数流这一概念的理解,还可应用到简单图的处处无零整数流的研究中。

[Purposes]Investigate the relationship between the existence of nowhere-zero integer flows of a multigraph and its base graph.[Methods]Use nowhere-zero integer flows of original graphs to construct nowhere-zero integer flows of new graphs.[Findings]There are two main results:1)If the base graph of a multigraph has a nowhere-zero k-flow,then it also has a nowhere-zero k-flow.2)If two multigraphs have a common base graph and each of their common edges is a parallel edge of an edge in the base graph,then the existence of nowhere-zero k-flows of the two multigraphs is identical.[Conclusions]The main results are not only helpful to understand the concept of nowhere-zero integer flow,but also can be applied to the study of nowhere-zero integer flows of simple graphs.