On Maximal Chain Subgraphs and Covers of Bipartite Graphs

We address the problem of enumeration of seating arrangements of married couples around a circular table such that no spouses seat next to each other and no k consecutive persons are of the same gender. While the case of k = 2 corresponds to the classical problème des ménages with a well-studied solution, no closed-form expression for number of arrangements is known when k ≥ 3. We propose a novel approach to this type of problems based on enumeration of circuits in certain algebraically weighted de Bruijn graphs. Our approach leads to a new expression for the menage numbers and their exponential generating function, and allows one to efficiently compute the number of seating arrangements in general cases. We work out all the details for k = 3.