Nowhere-zero Unoriented Flows in Hamiltonian Graphs

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 all of whose edges are extraverted. A nowhere-zero unoriented k-flow is an unoriented flow with values from the set {±1, . . . ,±(k − 1)}. It has been conjectured that if a graph has a nowhere-zero unoriented flow, then it admits a nowhere-zero unoriented 6-flow. We prove that this conjecture is true for hamiltonian graphs, with 6 replaced by 12. ∗