Let G be a connected bipartite graph with color classes E and V and root polytope Q. Regarding the hypergraph (V,E) induced by G, we prove that its interior polynomial is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It follows that the interior polynomials of (V,E) and its transpose (E,V) agree.
When G is a complete bipartite graph, our result recovers a well known hypergeometric identity due to Saalschütz. It also implies that certain extremal coefficients in the Homfly polynomial of a special alternating link can be read off of an associated Floer homology group.
Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun's conjecture, which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to two one-variable deformations of Cayley polytopes (which we call t-Cayley and t-Gayley polytopes), and to the most general two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph. Our approach is based on an explicit triangulation …
A common generalization for the chromatic polynomial and the flow polynomial of a graph $G$ is the Tutte polynomial $T(G;x,y)$. The combinatorial meaning for the coefficients of $T$ was discovered by Tutte at the beginning of its definition. However, for a long time the combinatorial meaning for the values of $T$ is missing, except for a few values such as $T(G;i,j)$, where $1\leq i,j\leq 2$, until recently for $T(G;1,0)$ and $T(G;0,1)$. In this third one of a series of papers, we introduce product valuations, cartesian product arrangements, and multivariable characteristic polynomials, and ap…
Tutte's dichromate T(x,y) is a well known graph invariant. Using the original definition in terms of internal and external activities as our point of departure, we generalize the valuations T(x,1) and T(1,y) to hypergraphs. In the definition, we associate activities to hypertrees, which are generalizations of the indicator function of the edge set of a spanning tree. We prove that hypertrees form a lattice polytope which is the set of bases in a polymatroid. In fact, we extend our invariants to integer polymatroids as well. We also examine hypergraphs that can be represented by planar bipartit…
This is a chapter destined for the book "Handbook of the Tutte Polynomial". The chapter is a composite. The first part is a brief introduction to Orlik-Solomon algebras. The second part sketches the theory of evaluative functions on matroid base polytopes and in particular, the G-invariant (as the subject is known late 2015). A third very short section is on Hopf-algebra or coalgebra structures in Tutte polynomial theory.
We recover the Tutte polynomial of a matroid, up to change of coordinates, from an Ehrhart-style polynomial counting lattice points in the Minkowski sum of its base polytope and scalings of simplices. Our polynomial has coefficients of alternating sign with a combinatorial interpretation closely tied to the Dawson partition. Our definition extends in a straightforward way to polymatroids, and in this setting our polynomial has Kálmán's internal and external activity polynomials as its univariate specialisations.
Tutte showed that a graph $G$ is planar if and only if the conflict graph associated to every cycle of $G$ is bipartite. We define a (not necessarily unique) signed conflict graph associated to a maximally planar subgraph of a nonplanar graph such that if $G$ has a flat embedding, every possible conflict graph associated to every maximally planar subgraph of $G$ is balanced. In doing this, we show that for every graph $G$ with flat embedding, and a planar subgraph $P$ of $G$, $P$ lies on a sphere that intersects $G$ only in $P$. We conjecture that $G$ is intrinsically linked if and only if eve…
The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a co…
We introduce a modular (integral) complementary polynomial $κ(G;x,y)$ ($κ_{\mathbbm z}(G;x,y)$) of two variables of a graph $G$ by counting the number of modular (integral) complementary tension-flows (CTF) of $G$ with an orientation $ε$. We study these polynomials by further introducing a cut-Eulerian equivalence relation on orientations and geometric structures such as the complementary open lattice polyhedron $Δ_\textsc{ctf}(G,ε)$, the complementary open 0-1 polytope $Δ^+_\textsc{ctf}(G,ε)$, and the complementary open lattice polytopes $Δ^ρ_\textsc{ctf}(G,ε)$ with respect to orientations $ρ…
Beck et. al. characterized the grid graphs whose perfect matching polytopes are Gorenstein and they also showed that for some parameters, perfect matching polytopes of torus graphs are Gorenstein. In this paper, we complement their result, that is, we characterize the torus graphs whose perfect matching polytopes are Gorenstein. Beck et. al. also gave a method to construct an infinite family of Gorenstein polytopes. In this paper, we introduce a new class of polytopes obtained from graphs and we extend their method to construct many more Gorenstein polytopes.
Given an oriented graph G, the modular flow polynomial counts the number of nowhere-zero Z_k-flows of G. We give a description of the modular flow polynomial in terms of (open) Ehrhart polynomials of lattice polytopes. Using Ehrhart-Macdonald reciprocity we give a combinatorial interpretation for the values of the modular flow polynomial at negative arguments which answers a question of Beck and Zaslavsky (2006). Our construction extends to Z_l-tensions and we recover Stanley's reciprocity theorem for the chromatic polynomial. Combining the combinatorial reciprocity statements for flows and te…
The Total Matching Polytope generalizes the Stable Set Polytope and the Matching Polytope. In this paper, we give the perfect formulation for Trees and we derive two new families of valid inequalities, the balanced biclique inequalities which are always facet-defining and the non-balanced lifted biclique inequalities obtained by a lifting procedure, which are facet-defining for bipartite graphs. Finally, we give a complete description for Complete Bipartite Graphs.
This paper presents an algorithm to construct a weighted adjacency matrix of a plane bipartite graph obtained from a pretzel knot diagram. The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel knot by way of perfect matchings (or dimers) of this graph. The weights are Tutte's activity letters that arise because the Jones polynomial is a specialization of the signed version of the Tutte polynomial. The relationship is formalized between the familiar spanning tree setting for the Tait graph and the perfect matchings of the plane bipartite graph above.…
The main result of this paper is that the polytope of the bipartite TSP is significantly different from that of the general TSP. Comb inequalities are known as facet defining ones in the general case. In the bipartite case, however, many of them are satisfied whenever all degree and subtour elimination constraints are satisfied, {\em i.e.}\ these comb inequalities are not facet defining. The inequalities in question belong to the cases where vertices of one of the two classes occur in less than the half of the intersections of the teeth and the hand. Such side conditions are necessary, as simp…
We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of all matchings, extended by a binary entry indicating whether the matching contains two specific edges. These polytopes are associated to the quadratic matching problems with a single linearized quadratic term. We provide a complete irredundant inequality description, which settles a conjecture by Klein (Ph.D. thesis, TU Dortmund, 2015). In addition, we also derive facetness and separation results for the polytopes. The completeness proof is based on a geometric relationship to a matching polytope of a non…
T. Kálmán (A version of Tutte's polynomial for hypergraphs, Adv. Math. 244 (2013) 823-873.) introduced the interior and exterior polynomials which are generalizations of the Tutte polynomial $T(x,y)$ on plane points $(1/x,1)$ and $(1,1/y)$ to hypergraphs. The two polynomials are defined under a fixed ordering of hyperedges, and are proved to be independent of the ordering using techniques of polytopes. In this paper, similar to the Tutte's original proof we provide a direct and elementary proof for the well-definedness of the interior and exterior polynomials of hypergraphs.
The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors, evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial. Here, we prove that the one-variable vertex-nullity interlace polynomial is in general #P-hard to compute. We also show a relation between the two-variable interlace polynomial and the topological Tut…
It is folklore that the cycle space of graphs of polytopes is generated by the cycles bounding the 2-faces. We provide a proof of this result that bypass homological arguments, which seem to be the most widely known proof. As a corollary, we obtain a result of Blind & Blind (1994) stating that graphs of polytopes are bipartite if and only if graphs of every 2-face are bipartite.
We introduce and study filtrations of a matroid on a linearly ordered ground set, which are particular sequences of nested sets. A given basis can be decomposed into a uniquely defined sequence of bases of minors, such that these bases have an internal/external activity equal to 1/0 or 0/1 (in the sense of Tutte polynomial activities). This decomposition, which we call the active filtration/partition of the basis, refines the known partition of the ground set into internal and external elements with respect to a given basis. It can be built by a certain closure operator, which we call the acti…
The Loopy Polynomial: from Tutte's Universal $V$-Function to Bizonotopal Geometry
Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（燕京数学科学应用研究院）；Stockholm University（斯德哥尔摩大学）；Guangdong Technion–Israel Institute of Technology（广东以色列理工学院）；University of Oregon（俄勒冈大学）
This is the first one of a series of papers on association of orientations, lattice polytopes, and abelian group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the…
The extension complexity $\mathsf{xc}(P)$ of a polytope $P$ is the minimum number of facets of a polytope that affinely projects to $P$. Let $G$ be a bipartite graph with $n$ vertices, $m$ edges, and no isolated vertices. Let $\mathsf{STAB}(G)$ be the convex hull of the stable sets of $G$. It is easy to see that $n \leqslant \mathsf{xc} (\mathsf{STAB}(G)) \leqslant n+m$. We improve both of these bounds. For the upper bound, we show that $\mathsf{xc} (\mathsf{STAB}(G))$ is $O(\frac{n^2}{\log n})$, which is an improvement when $G$ has quadratically many edges. For the lower bound, we prove that …
We consider the problem of constructing an abstract $(n+1)$-polytope $Q$ with $k$ facets isomorphic to a given $n$-polytope $P$, where $k \geq 3$. In particular, we consider the case where we want $Q$ to be $(n-2,n)$-flat, meaning that every $(n-2)$-face is incident to every $n$-face (facet). We show that if $P$ admits such a flat extension for a given $k$, then the facet graph of $P$ is $(k-1)$-colorable. Conversely, we show that if the facet graph is $(k-1)$-colorable and $k-1$ is prime, then $P$ admits a flat extension for that $k$. We also show that if $P$ is facet-bipartite, then for ever…
In this paper, we introduce polytopes ${\mathcal B}_G$ arising from root systems $B_n$ and finite graphs $G$, and study their combinatorial and algebraic properties. In particular, it is shown that ${\mathcal B}_G$ is reflexive if and only if $G$ is bipartite. Moreover, in the case, ${\mathcal B}_G$ has a regular unimodular triangulation. This implies that the $h^*$-polynomial of ${\mathcal B}_G$ is palindromic and unimodal when $G$ is bipartite. Furthermore, we discuss stronger properties, namely the $γ$-positivity and the real-rootedness of the $h^*$-polynomials. In fact, if $G$ is bipartite…
A graph is called $k$-extendable if each $k$-matching can be extended to a perfect matching. We give spectral conditions for the $k$-extendability of graphs and bipartite graphs using Tutte-type and Hall-type structural characterizations. Concretely, we give a sufficient condition in terms of the spectral radius of the distance matrix for the $k$-extendability of a graph and completely characterize the corresponding extremal graphs. A similar result is obtained for bipartite graphs.
The interior polynomial is an invariant of (signed) bipartite graphs, and the interior polynomial of a plane bipartite graph is equal to a part of the HOMFLY polynomial of a naturally associated link. The HOMFLY polynomial $P_L(v,z)$ is a famous link invariant with many known properties. For example, the HOMFLY polynomial of the mirror image of $L$ is given by $P_{L}(-v^{-1},z)$. This implies a property of the interior polynomial in the planar case. We prove that the same property holds for any bipartite graph. The proof relies on Ehrhart reciprocity applied to the so called root polytope. We …
The cut polytope of a graph arises in many fields. Although much is known about facets of the cut polytope of the complete graph, very little is known for general graphs. The study of Bell inequalities in quantum information science requires knowledge of the facets of the cut polytope of the complete bipartite graph or, more generally, the complete k-partite graph. Lifting is a central tool to prove certain inequalities are facet inducing for the cut polytope. In this paper we introduce a lifting operation, named triangular elimination, applicable to the cut polytope of a wide range of graphs.…
We consider the computational problem of finding short paths in the skeleton of the perfect matching polytope of a bipartite graph. We prove that unless $P=NP$, there is no polynomial-time algorithm that computes a path of constant length between two vertices at distance two of the perfect matching polytope of a bipartite graph. Conditioned on $P\neq NP$, this disproves a conjecture by Ito, Kakimura, Kamiyama, Kobayashi and Okamoto [SIAM Journal on Discrete Mathematics, 36(2), pp. 1102-1123 (2022)]. Assuming the Exponential Time Hypothesis we prove the stronger result that there exists no poly…
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees on the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to ``discrete analytic functions'' on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we class…
Symmetric edge polytopes, a.k.a. PV-type adjacency polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular, the authors are motivated by the study of the algebraic Kuramoto equations of unmixed form whose Newton polytopes are the symmetric edge polytopes. The interplay between the geometric structure of symmetric edge polytopes and the topological structure of the underlying graphs has been a recurring theme in recent studies. In particular, ``facet/face su…
The cut polytope of a graph is an important object in several fields, such as functional analysis, combinatorial optimization, and probability. For example, Sturmfels and Sullivant showed that the toric ideals of cut polytopes are useful in algebraic statistics. In the theory of lattice polytopes, the $h^{*}$-polynomial is one of the most important invariants. The necessary and sufficient condition in terms of graphs that the $h^{*}$-polynomial of a cut polytope is palindromic is known. However, except for trees, there are no classes of graphs for which the $h^{*}$-polynomial of their cut poly…
We show that computing the Tutte polynomial of a linear matroid of dimension $k$ on $k^{O(1)}$ points over a field of $k^{O(1)}$ elements requires $k^{Ω(k)}$ time unless the \#ETH---a counting extension of the Exponential Time Hypothesis of Impagliazzo and Paturi [CCC 1999] due to Dell {\em et al.} [ACM TALG 2014]---is false. This holds also for linear matroids that admit a representation where every point is associated to a vector with at most two nonzero coordinates. We also show that the same is true for computing the Tutte polynomial of a binary matroid of dimension $k$ on $k^{O(1)}$ point…
Geometric Realizations with Strong Self-Duality Part I: Non-Bipartite Quadrangulations of the Projective Plane and Strongly Involutive Self-Dual Graphs
HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究网络阿尔弗雷德·雷尼数学研究所）；ELTE Eötvös Loránd University（厄特沃什·罗兰大学）
In this paper we study lift-and-project polyhedral operators defined by Lov?asz and Schrijver and Balas, Ceria and Cornu?ejols on the clique relaxation of the stable set polytope of web graphs. We compute the disjunctive rank of all webs and consequently of antiweb graphs. We also obtain the disjunctive rank of the antiweb constraints for which the complexity of the separation problem is still unknown. Finally, we use our results to provide bounds of the disjunctive rank of larger classes of graphs as joined a-perfect graphs, where near-bipartite graphs belong.
We present a generalization of the marriage problem underlying Hall's famous Marriage Theorem to what we call the Symmetric Marriage Problem, a problem that can be thought of as a special case of Maximal Weighted Bipartite Matching. We show that there is a solution to the Symmetric Marriage Problem if and only if a variation on Hall's Condition holds on each of the bipartitions. We prove both finite and infinite versions of this result and provide applications. We also introduce a non-bipartite version of the problem and show that a generalization of Tutte's Theorem applies.
We provide an upper bound to the number of graph homomorphisms from $G$ to $H$, where $H$ is a fixed graph with certain properties, and $G$ varies over all $N$-vertex, $d$-regular graphs. This result generalizes a recently resolved conjecture of Alon and Kahn on the number of independent sets. We build on the work of Galvin and Tetali, who studied the number of graph homomorphisms from $G$ to $H$ when $H$ is bipartite. We also apply our techniques to graph colorings and stable set polytopes.
An integral convex polytope ${\mathcal P}$ is said to be Gorenstein if its toric ring $K[{\mathcal P}]$ is normal and Gorenstein. In this paper, Gorenstein cut polytopes of graphs are characterized explicitly. First, we prove that Gorenstein cut polytopes are compressed (i.e., all of whose reverse lexicographic triangulations are unimodular). Second, by applying Athanasiadis's theory for Gorenstein compressed polytopes, we show that a cut polytope of a graph $G$ is Gorenstein if and only if $G$ has no $K_5$-minor and $G$ is either a bipartite graph without induced cycles of length $\geq 6$ or …
The 2-block intersection graph (2-BIG) of a twofold triple system (TTS) is the graph whose vertex set is composed of the blocks of the TTS and two vertices are joined by an edge if the corresponding blocks intersect in exactly two elements. The 2-BIGs are themselves interesting graphs: each component is cubic and 3-connected, and a 2-BIG is bipartite exactly when the TTS is decomposable to two Steiner triple systems. Any connected bipartite 2-BIG with no Hamilton cycle is a counter-example to a conjecture posed by Tutte in 1971. Our main result is that there exists an integer $N$ such that for…
Hibi showed that the polynomial in the numerator of the Ehrhart series of a reflexive polytope is palindromic. We proved that those in the numerator of the Ehrhart series of every graph polytope (defined later) of the bipartite graph is palindromic. From this, one of the conjectures (raised in the A205497 of OEIS \cite{[O]}) follows immediately.
Recently O. Bernardi gave a formula for the Tutte polynomial $T(x,y)$ of a graph, based on spanning trees and activities just like the original definition, but using a fixed ribbon structure to order the set of edges in a different way for each tree. The interior polynomial $I$ is a generalization of $T(x,1)$ to hypergraphs. We supply a Bernardi-type description of $I$ using a ribbon structure on the underlying bipartite graph $G$. Our formula works because it is determined by the Ehrhart polynomial of the root polytope of $G$ in the same way as $I$ is. To prove this we interpret the Bernardi …
The Ehrhart ring of the edge polytope $\mathcal{P}_G$ for a connected simple graph $G$ is known to coincide with the edge ring of the same graph if $G$ satisfies the odd cycle condition. This paper gives for a graph which does not satisfy the condition, a generating set of the defining ideal of the Ehrhart ring of the edge polytope, described by combinatorial information of the graph. From this result, two factoring properties of the Ehrhart series are obtained; the first one factors out bipartite biconnected components, and the second one factors out a even cycle which shares only one edge wi…
We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the $h^\ast$-polynomial in case of complete bipartite graphs. In particular, we show that the $h^\ast$-polynomial is $γ$-positive and real-rooted. This proves Gal's conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthing due to Nevo and Petersen (20…
The interior polynomial and the exterior polynomial are generalizations of valuations on $(1/ξ,1)$ and $(1,1/η)$ of the Tutte polynomial $T_G(x,y)$ of graphs to hypergraphs, respectively. The pair of hypergraphs induced by a connected bipartite graph are abstract duals and are proved to have the same interior polynomial, but may have different exterior polynomials. The top of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the pair of hypergraphs induced by the Seifert graph of the link. Let $G=(V\cup E, \varepsilon)$ be a connected bipartite graph…
Let $d \geq 3$ be an integer. It is known that the number of edges of the edge polytope of the complete graph with $d$ vertices is $d(d-1)(d-2)/2$. In this paper, we study the maximum possible number $μ_d$ of edges of the edge polytope arising from finite simple graphs with $d$ vertices. We show that $μ_{d}=d(d-1)(d-2)/2$ if and only if $3 \leq d \leq 14$. In addition, we study the asymptotic behavior of $μ_d$. Tran--Ziegler gave a lower bound for $μ_d$ by constructing a random graph. We succeeded in improving this bound by constructing both a non-random graph and a random graph whose compleme…
In 1971, Tutte wrote in an article that "it is tempting to conjecture that every 3-connected bipartite cubic graph is hamiltonian". Motivated by this remark, Horton constructed a counterexample on 96 vertices. In a sequence of articles by different authors several smaller counterexamples were presented. The smallest of these graphs is a graph on 50 vertices which was discovered independently by Georges and Kelmans. In this article we show that there is no smaller counterexample. As all non-hamiltonian 3-connected bipartite cubic graphs in the literature have cyclic 4-cuts -- even if they have …
