Nowhere-zero Unoriented Flows in Hamiltonian Graphs

Abstract

An unoriented flow in a graph is an assignment of real numbers to the edges such that the sum of the values of all edges incident with each vertex is zero. This is equivalent to a flow in a bidirected graph where all edges are extraverted. A nowhere-zero unoriented \(k\)-flow is an unoriented flow with values from the set \(\{\pm 1, \ldots, \pm( k-1)\}\). It has been conjectured that if a graph admits a nowhere-zero unoriented flow, then it also admits a nowhere-zero unoriented \(6\)-flow. We prove that this conjecture holds true for Hamiltonian graphs, with \(6\) replaced by \(12\).