# Zero-Sum 6-Flows in 5-Regular Graphs

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## Abstract

Let *G* be a graph. A zero-sum flow of *G* is an assignment of nonzero real numbers to the edges of *G* such that the sum of the values of all edges incident with each vertex is zero. Let *k* be a natural number. A zero-sum *k*-flow is a flow with values from the set \(\{\pm 1, \ldots , \pm (k - 1)\}\). In this paper, we prove that every 5-regular graph admits a zero-sum 6-flow.

## Keywords

Zero-sum flow 5-Regular Bidirected graph Cycle-cubic tree## Notes

### Acknowledgements

The first author appreciates Professor Sanming Zhou for his valuable comments and advisements. Moreover, the first author acknowledges the hospitality of School of Mathematics and Statistics, The University of Melbourne during her visit while the research on this work was conducted. The authors would like to express their gratitude to the referee for her/his careful reading and helpful comments.

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