  Double Hurwitz numbers have at least four equivalent definitions. Most
naturally, they count covers of the Riemann sphere by genus g curves with
certain specified ramification data. This is classically equivalent to counting
certain collections of permutations. More recently, double Hurwitz numbers have
been expressed as a count of certain ribbon graphs, or as a weighted count of
certain labeled graphs.
  This note is an expository account of the equivalences between these
definitions, with a few novelties. In particular, we give a simple
combinatorial algorithm to pass directly between the permutation and ribbon
graph definitions. The two graph theoretic points of view have been used to
give proofs that double Hurwitz numbers are piecewise polynomial. We use our
algorithm to compare these two proofs.
