Enumeration of paths and cycles and e-coefficients of incomparability graphs

We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in terms of elementary symmetric functions. Analysis of some of the combinatorial implications of this expansion leads to three bijections involving acyclic orientations

Tutte Short Exact Sequences of Graphs

We associate two modules, the $G$-parking critical module and the toppling critical module, to an undirected connected graph $G$. We establish a Tutte-like short exact sequence relating the modules associated to $G$, an edge contraction $G/e$ and edge deletion $G \setminus e$ ($e$ is a non-bridge). As applications of these short exact sequences, we relate the vanishing of certain combinatorial invariants (the number of acyclic orientations on connected partition graphs satisfying a unique sink property) of $G/e$ to the equality of corresponding invariants of $G$ and $G \setminus e$. We also obtain a short proof of a theorem of Merino that the critical polynomial of a graph is an evaluation of its Tutte polynomial.Comment: 40 pages, 3 figure