Nowhere-Zero Flows in Random Graphs

A nowhere-zero 3-flow in a graph G is an assignment of a direction and a value of 1 or 2 to each edge of G such that, for each vertex v in G, the sum of the values of the edges with tail v equals the sum of the values of the edges with head v. Motivated by results about the region coloring of planar graphs, Tutte conjectured in 1966 that every 4-edge-connected graph has a nowhere-zero 3-flow. This remains open. In this paper we study nowhere-zero flows in random graphs and prove that almost surely as soon as the random graph G(n, p) has minimum degree two it has a nowhere-zero 3-flow. This result is clearly best possible.