Partitions of an Eulerian Digraph into Circuits

TL;DR

This paper proves a cancellation property for Eulerian digraphs involving partitions into circuits, linking Martin polynomials with chromatic polynomials, and applies it to classical graph theorems.

Contribution

It introduces an alternative proof of the cancellation property using Heaps of Pieces and bijections, connecting Martin polynomials with intersection graph chromatic polynomials.

Findings

Proves a sum cancellation property for Eulerian digraph partitions.

Establishes a bijection between trails and heaps with a unique maximal piece.

Relates Martin polynomial to chromatic polynomials of intersection graphs.

Abstract

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.