Zero-Sum Flows in Regular Graphs
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- 作者：S. Akbari ; A. Daemi ; O. Hatami ; A. Javanmard and A. Mehrabian
- 关键词：Regular graph ; Bidirected graph ; Zero ; sum flow
- 刊名：Graphs and Combinatorics
- 出版时间：September 2010
- 出版年：2010
- 期刊代码：63_0911-0119
- 类别：mat
- 卷：26
- 期：5
- 页码：603-615
- 数据来源：sp
摘要
For an undirected graph G, a zero-sum flow is an assignment of non-zero real numbers to the edges, such that the sum of the values of all edges incident with each vertex is zero. It has been conjectured that if a graph G has a zero-sum flow, then it has a zero-sum 6-flow. We prove this conjecture and Bouchet’s Conjecture for bidirected graphs are equivalent. Among other results it is shown that if G is an r-regular graph (r ≥ 3), then G has a zero-sum 7-flow. Furthermore, if r is divisible by 3, then G has a zero-sum 5-flow. We also show a graph of order n with a zero-sum flow has a zero-sum (n + 3)2-flow. Finally, the existence of k-flows for small graphs is investigated.