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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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