Partition functions and a generalized coloring‐flow duality for embedded graphs

Abstract

AbstractLet G be a finite group and a class function. Let be a directed graph with for each vertex a cyclic order of the edges incident to it. The cyclic orders give a collection F of faces of H. Define the partition function , where denotes the product of the κ‐values of the edges incident with v (in cyclic order), where the inverse is taken for edges leaving v. Write , where the sum runs over irreducible representations λ of G with character and with for every λ. When H is connected, it is proved that , where 1 is the identity element of G. Among the corollaries, a formula for the number of nowhere‐identity G‐flows on H is derived, generalizing a result of Tutte. We show that these flows correspond bijectively to certain proper G‐colorings of a covering graph of the dual graph of H. This correspondence generalizes coloring‐flow duality for planar graphs.