A note on zero-sum 5-flows in regular graphs (Q426866)

Summary: Let \(G\) be a graph. A zero-sum flow of \(G\) is an assignment of non-zero 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 \(\{\pm 1,\ldots ,\pm(k-1)\}\). It has been conjectured that every \(r\)-regular graph, \(r\geq 3\), admits a zero-sum 5-flow. In this paper we provide an affirmative answer to this conjecture, except for \(r=5\).