Graphs and Combinatorics

, Volume 26, Issue 5, pp 603–615 | Cite as

Zero-Sum Flows in Regular Graphs

  • S. AkbariEmail author
  • A. Daemi
  • O. Hatami
  • A. Javanmard
  • A. Mehrabian
Original Paper


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.


Regular graph Bidirected graph Zero-sum flow 

Mathematics Subject Classification (2000)

05C21 05C50 


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  1. 1.
    Akbari S., Ghareghani N., Khosrovshahi G.B., Mahmoody A.: On zero-sum 6-flows of graphs. Linear Algebra Appl. 430, 3047–3052 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alon N.: Combinatorial Nullstellensatz, Recent trends in combinatorics. Combin. Probab. Comput. 8(1–2), 7–29 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bouchet A.: Nowhere-zero integral flows on a bidirected graph. J. Combin. Theory Ser. B 34, 279–292 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Graham R.L., Grötschel M., Lovász L.: Handbook of Combinatorics. The MIT Press/North-Holland, Amsterdam (1995)zbMATHGoogle Scholar
  5. 5.
    Hall P.: On representatives of subsets. J. Lond. Math. Soc. 10, 26–30 (1935)zbMATHCrossRefGoogle Scholar
  6. 6.
    Jaeger F.: Flows and generalized coloring theorems in graphs. J. Combin. Theory Ser. B 26(2), 205–216 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Seymour P.D.: Nowhere-zero 6-flows. J. Combin. Theory Ser. B 30(2), 130–135 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Tutte W.T.: A contribution to the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954)zbMATHMathSciNetGoogle Scholar
  9. 9.
    West D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Englewood Cliffs (2001)Google Scholar
  10. 10.
    Xu R., Zhang C.: On flows in bidirected graphs. Discrete Math. 299, 335–343 (2005)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2010

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

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