Jo Ellis-Monaghan (University of Vermont)
The Martin polynomials encode information about families of closed paths in four-regular Eulerian graphs and digraphs. The circuit partition polynomials transform the Martin polynomials into particularly easy to manipulate forms and generalize them to all Eulerian graphs. New identities for these polynomials, analogous to Tutte's identity for the chromatic polynomial, give expressions that rapidly expand a very small handful of evaluations into a wealth of new information. An identity of Martin and Las Vergnas uses medial graphs to relate the circuit partition polynomials to the Tutte polynomial of planar graphs along the diagonal y = x. Previously, interpretations were known for only two non-trivial evaluations on this line. Now however, the new identities for the circuit partition polynomials give combinatorial interpretations for all integer values of the diagonal Tutte polynomial of a planar graph.