It is pointed out that a graph contributes to the high-temperature expansion of the M -state Potts model if and only if it has a nowhere-zero M -flow. This graph-theoretical property is shown to be equivalent to the existence of a proper arrowing of the graph, which can be visualized easily by means of arrows put on the edges in such a way that the sum of these arrows is 0 mod M at every vertex. By means of this concept it is rather easy to solve the problem for planar graphs completely, one result being that all planar graphs contribute for M 4. For nonplanar graphs, all of them certainly contribute for M 6, but it is an unproven conjecture that M 5 suffices. The classes of graphs contributing for 2 M 6 are characterized.