Graphs and Combinatorics

, Volume 26, Issue 5, pp 603–615

# Zero-Sum Flows in Regular Graphs

Original Paper

## Abstract

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.

## Keywords

Regular graph Bidirected graph Zero-sum flow

05C21 05C50

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## Authors and Affiliations

• S. Akbari
• 1
• 5
Email author
• A. Daemi
• 2
• O. Hatami
• 1
• A. Javanmard
• 3
• A. Mehrabian
• 4
1. 1.Department of Mathematical SciencesSharif University of TechnologyTehranIran
2. 2.Department of MathematicsHarvard UniversityCambridgeUSA
3. 3.Department of Electrical EngineeringStanford UniversityStanfordUSA
4. 4.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
5. 5.School of Mathematics, Institute for Research in Fundamental Sciences (IPM)TehranIran