Even subgraph expansions for the flow polynomial of cubic plane maps (Q1823252)

We give two new expansions of the flow polynomial \(F(G,\lambda)\) of a cubic plane map \(G\). The first expansion is in terms of oriented even subgraphs of \(G\), and the second one is an unoriented version of the first. These expansions depend on topological properties of the plane embedding and cannot in general be extended to non-planar graphs. There are two exceptions: for \(\lambda =4\) our formulas degenerate to a triviality, and for \(\lambda =0\) we obtain a simple expansion valid for general cubic graphs. The Beraha numbers appear naturally as special values for our expansions.