Zero-Sum Flow Numbers of Regular Graphs

  • Tao-Ming Wang
  • Shih-Wei Hu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7285)

Abstract

As an analogous concept of a nowhere-zero flow for directed graphs, we consider zero-sum flows for undirected graphs in this article. For an undirected graph G, a zero-sum flow is an assignment of non-zero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum k -flow if the values of edges are less than k. We define the zero-sum flow number of G as the least integer k for which G admitting a zero-sum k-flow. In this paper, among others we calculate the zero-sum flow numbers for regular graphs and also the zero-sum flow numbers for Cartesian products of regular graphs with paths.

Keywords

Directed Graph Perfect Match Undirected Graph Regular Graph Edge Label 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Akbari, S., Daemi, A., Hatami, O., Javanmard, A., Mehrabian, A.: Zero-Sum Flows in Regular Graphs. Graphs and Combinatorics 26, 603–615 (2010)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Akbari, S., Ghareghani, N., Khosrovshahi, G.B., Mahmoody, A.: On zero-sum 6-flows of graphs. Linear Algebra Appl. 430, 3047–3052 (2009)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Akbari, S., et al.: A note on zero-sum 5-flows in regular graphs. arXiv:1108.2950v1 [math.CO] (2011)Google Scholar
  4. 4.
    Bouchet, A.: Nowhere-zero integral flows on a bidirected graph. J. Combin. Theory Ser. B 34, 279–292 (1983)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Gallai, T.: On factorisation of grahs. Acta Math. Acad. Sci. Hung 1, 133–153 (1950)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Jaeger, F.: Flows and generalized coloring theorems in graphs. J. Combin. Theory Ser. B 26(2), 205–216 (1979)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Kano, M.: Factors of regular graph. J. Combin. Theory Ser. B 41, 27–36 (1986)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Petersen, J.: Die Theorie der regularen graphs. Acta Mathematica (15), 193–220 (1891)Google Scholar
  9. 9.
    Seymour, P.D.: Nowhere-zero 6-flows. J. Combin. Theory Ser. B 30(2), 130–135 (1981)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Tutte, W.T.: A contribution to the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Wang, T.-M., Hu, S.-W.: Constant Sum Flows in Regular Graphs. In: Atallah, M., Li, X.-Y., Zhu, B. (eds.) FAW-AAIM 2011. LNCS, vol. 6681, pp. 168–175. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Wang, T.-M., Hu, S.-W.: Nowhere-zero constant-sum flows of graphs. Presented in the 2nd India-Taiwan Conference on Discrete Mathematics, Coimbatore, Tamil Nadu, India (September 2011) (manuscript)Google Scholar
  13. 13.
    West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Englewood Cliffs (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Tao-Ming Wang
    • 1
  • Shih-Wei Hu
    • 1
  1. 1.Department of Applied MathematicsTunghai UniversityTaichungR.O.C.

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