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Germain Poullot

Publications and source records attributed to Germain Poullot.

11 recordsLinked to original sources
arXiv2026

Hamiltonicity of graphs of acyclic orientations and acyclic polynomials

We study the graph $\mathcal{AO}(G)$ of acyclic orientations of a graph $G$. Two acyclic orientations are adjacent in this graph if they disagree on the orientation of a single arc. In particular, we focus on the Hamiltonicity of the graphs $\mathcal{AO}(G)$. Using two methods of pattern lacing which generalize the zig-zag method of Brenner, Cardinal, McConville, Merino and M\"utze, we characterize which multipaths are $\mathcal{AO}$-Hamiltonian. Moreover, we give a criterion for the gluing of a multipath on a given graph to preserve $\mathcal{AO}$-Hamiltonicity. Building towards an inductive certification of $\mathcal{AO}$-Hamiltonicity via the (open) ear decomposition of 2-connected graphs, we propose three ways of gluing several multipaths to a given graph. In addition, we define the acyclic polynomials to encapsulate both the number of acyclic orientations of a graph and the "parity problem" proposed by Savage, Squire and West: if $-1$ is not a root of the acyclic polynomial of $G$, then $G$ is not $\mathcal{AO}$-Hamiltonian. We explore numerous properties of the acyclic polynomials, proving that they are not instances of the famous Tutte-Whitney polynomials, but that they too exhibit a partial deletion-contraction phenomenon.

math.CO↗
arXiv2026

Interval hypergraphic polytopes (or deformed associahedra), Tamari interval posets, and weeping willows

For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic polytope $\triangle_{\mathbb{H}}$ is the Minkowski sum of the standard simplices $\triangle_H$ for all $H \in \mathbb{H}$. We focus here on interval hypergraphs, where all hyperedges are intervals of $[n]$. They are precisely the deformations of Loday's associahedron. Their vertex posets are Tamari interval posets, and we describe which Tamari interval poset appears as a vertex poset in which interval hypergraphic polytope. We also characterize the interval hypergraphs $\mathbb{I}$ for which the hypergraphic polytope $\triangle_\mathbb{I}$ is simple, and we study their vertex posets, which we call weeping willows.

math.CO↗
arXiv2025

The graph of implicit edge dependencies for indecomposability and beyond

A polytope is called indecomposable if it cannot be expressed nontrivially as a Minkowski sum of other polytopes. Since Gale introduced the concept in 1954, several increasingly strong criteria have been developed to characterize indecomposability. In this paper, we introduce a new approach to indecomposability for frameworks and polytopes based on the graph of implicit edge dependencies, which records proportionalities between edge lengths across all deformations. This yields a new indecomposability criterion that unifies and generalizes most previous approaches, and has additional consequences in the study of deformation cones. As a main application, we construct new indecomposable deformed permutahedra that are not matroid polytopes. In 1970, Edmonds already noted the difficulty of characterizing the extreme rays of the submodular cone, equivalently, indecomposable deformed permutahedra. Matroid polytopes of connected matroids form a well-known family of such examples. We exhibit a new infinite family of indecomposable deformations of the permutahedron, not arising from matroid polytopes, obtained by suitable truncations of certain graphical zonotopes. We further demonstrate the scope of our methods through several additional applications. In particular, we refute a conjecture of Smilansky (1987) on the relation between the numbers of vertices and facets of indecomposable polytopes. Moreover, we obtain new bounds on the dimensions of deformation cones and we construct and analyze uniquely decomposable polytopes.

math.CO↗
arXiv2025

Deformation cones of graphical zonotopes for $K_4$-free graph

In this paper, we compute a triangulation of certain faces of the submodular cone. More precisely, graphical zonotopes are generalized permutahedra, and hence their deformation cones are faces of the submodular cone. We give a triangulation of these faces for graphs without induced complete sub-graph on 4 vertices. We deduce the rays of these faces: Minkowski indecomposable deformations of these graphical zonotopes are segments and triangles. Besides, computer experiments lead to examples of graphs without induced complete sub-graph on 5 vertices, whose graphical zonotopes have high dimensional Minkowski indecomposable deformations.

math.CO↗
arXiv2025

Vertices of the monotone path polytopes of hypersimplicies

The monotone path polytope of a polytope $P$ encapsulates the combinatorial behavior of the shadow vertex rule (a pivot rule used in linear programming) on $P$. Computing monotone path polytopes is the entry door to the larger subject of fiber polytopes, for which explicitly computing examples remains a challenge. We first give a detailed presentation on how to construct monotone path polytopes. Monotone path polytopes of cubes and simplices have been known since the seminal article of Billera and Sturmfels. We extend these results to hypersimplices by linking this problem to the combinatorics of lattice paths. Indeed, we give a combinatorial model which describes the vertices of the monotone path polytope of the hypersimplex $Δ(n, 2)$ (for any generic direction). With this model, we give a precise count of these vertices, and furthermore count the number of coherent monotone paths on $Δ(n, 2)$ according to their lengths. We prove that some of the results obtained also hold for hypersimplices $Δ(n, k)$ for $k\geq 2$.

math.CO↗
arXiv2025

Many rays of the submodular cone

The study of the cone of submodular functions goes back to Jack Edmonds' seminal 1970 paper, which already highlighted the difficulty of characterizing its extreme rays. Since then, researchers from diverse fields have sought to characterize, enumerate, and bound the number of such rays. In this paper, we introduce an inductive construction that generates new rays of the submodular cone. This allows us to establish that the $n$-th submodular cone has at least $2^{2^{n-2}}$ rays, which improves upon the lower bound obtained from Hien Q. Nguyen's 1986 characterization of indecomposable matroid polytopes by a factor of order $\sqrt{n^3}$ in the exponent.

math.CO↗
arXiv2025

Ehrhart non-positivity and unimodular triangulations for classes of s-lecture hall simplices

Counting lattice points and triangulating polytopes is a prominent subject in discrete geometry, yet proving Ehrhart positivity or existence of unimodular triangulations remain of utmost difficulty in general, even for ``easy'' simplices. We study these questions for classes of s-lecture hall simplices. Inspired by a question of Olsen, we present a new natural class of sequences s for which the s-lecture hall simplices are not Ehrhart positive, by explicitly estimating a negative coefficient. Meanwhile, motivated by a conjecture of Hibi, Olsen and Tsuchiya, we extend the previously known classes of sequences s for which the s-lecture hall simplex admits a flag, regular and unimodular triangulation. The triangulations we construct are explicit.

math.CO↗
arXiv2025

Unimodality of the number of paths per length on polytopes: Examples, counterexamples, and a central limit theorem

Because of its importance in combinatorics and optimization we study the full distribution of the lengths of monotone paths of a convex polytope. De Loera had conjectured that the number of such paths counted by length is always a unimodal sequence. We confirm this for several classes of polytopes but construct counterexamples disproving the conjecture in general. Nevertheless, we show that for random polytopes with vertices uniformly distributed on the sphere, the length of coherent paths satisfies a central limit theorem.

math.CO↗
arXiv2022

Deformed graphical zonotopes

We study deformations of graphical zonotopes. Deformations of the classical permutahedron (which is the graphical zonotope of the complete graph) have been intensively studied in recent years under the name of generalized permutahedra. We provide an irredundant description of the deformation cone of the graphical zonotope associated to a graph $G$, consisting of independent equations defining its linear span (in terms of non-cliques of $G$) and of the inequalities defining its facets (in terms of common neighbors of neighbors in $G$). In particular, we deduce that the faces of the standard simplex corresponding to induced cliques in $G$ form a linear basis of the deformation cone, and that the deformation cone is simplicial if and only if $G$ is triangle-free.

math.CO↗
arXiv2022

Deformation cones of graph associahedra and nestohedra

We give the facet description of the deformation cones of graph associahedra and nestohedra, generalizing the classical parametrization of the family of deformed permutahedra by the cone of submodular functions. When the underlying building set is made of intervals, this leads in particular to the construction of kinematic nestohedra generalizing the kinematic associahedra that recently appeared in the theory of scattering amplitudes.

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