Partitions of an Eulerian Digraph into Circuits

We investigate a cancellation property satisfied by a connected Eulerian digraph $D$. Namely, unless $D$ is a single directed cycle, we have $\sum_{k\geq 1} (-1)^{k} f_k(D)=0$, where $f_k(D)$ is the number of partitions of Eulerian circuits of $D$ into $k$ circuits. This property is a consequence of the fact that the Martin polynomial of a digraph has no constant term. We provide an alternative proof by employing Viennot's theory of Heaps of Pieces, and in particular, a bijection between closed trails of a digraph and heaps with a unique maximal piece, which are also in bijection with unique sink orientations of the intersection graphs $G_a$ of partitions $a$ of $E(D)$ into cycles. The argument considers the partition lattice of the edge set of a digraph $D$, restricted to the join-semilattice $T(D)$ induced by elements whose blocks are connected and Eulerian. The minimal elements of $T(D)$ are exactly the partitions of $D$ into cycles, and the up-set of a minimal element $a\in T(D)$ is shown to be isomorphic to the bond lattice $L(G_a)$. Using tools developed by Whitney and Rota, we perform M\"{o}bius inversion on $T(D)$ and obtain the claimed cancellation. As a consequence of this alternative proof, we relate the Martin polynomial of a digraph directly to the chromatic polynomials of the intersection graphs of partitions of $D$ into cycles. Finally, we apply the cancellation property in order to deduce the classical Harary-Sachs Theorem for graphs of rank $2$ from a hypergraph generalization thereof, remedying a gap in a previous proof of this.