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Zero-Sum Flows in Regular Graphs

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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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

    S. Akbari & O. Hatami

  2. Department of Mathematics, Harvard University, Cambridge, USA

    A. Daemi

  3. Department of Electrical Engineering, Stanford University, Stanford, USA

    A. Javanmard

  4. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada

    A. Mehrabian

  5. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran

    S. Akbari

Authors
  1. S. Akbari
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  2. A. Daemi
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  3. O. Hatami
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  4. A. Javanmard
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  5. A. Mehrabian
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Correspondence to S. Akbari.

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Akbari, S., Daemi, A., Hatami, O. et al. Zero-Sum Flows in Regular Graphs. Graphs and Combinatorics 26, 603–615 (2010). https://doi.org/10.1007/s00373-010-0946-5

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  • DOI: https://doi.org/10.1007/s00373-010-0946-5

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