Satyan L. Devadoss, Mia Smith
Given a graph G, the graph associahedron is a simple convex polytope whose face poset is based on the connected subgraphs of G. With the additional assignment of a color palette, we define the colorful graph associahedron, show it to be a collection of simple abstract polytopes, and explore its properties.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Every n-edge colored n-regular graph G naturally gives rise to a simple abstract n-polytope, the colorful polytope of G, whose 1-skeleton is isomorphic to G. The paper describes colorful polytope versions of the associahedron and cyclohedron. Like their classical counterparts, the colorful associahedron and cyclohedron encode triangulations and flips, but now with the added feature that the diagonals of the triangulations are colored and adjacency of triangulations requires color preserving flips. The colorful associahedron and cyclohedron are derived as colorful polytopes from the edge colore…
A graph associahedron is a simple polytope whose face lattice encodes the nested structure of the connected subgraphs of a given graph. In this paper, we study certain graph properties of the 1-skeleta of graph associahedra, such as their diameter and their Hamiltonicity. Our results extend known results for the classical associahedra (path associahedra) and permutahedra (complete graph associahedra). We also discuss partial extensions to the family of nestohedra.
A graph associahedron is a polytope dual to a simplicial complex whose elements are induced connected subgraphs called tubes. Graph associahedra generalize permutahedra, associahedra, and cyclohedra, and therefore are of great interest to those who study Coxeter combinatorics. This thesis characterizes nested complexes of simplicial complexes, which we call $Δ$-nested complexes. From here, we can define P-nestohedra by truncating simple polyhedra, and in more specificity define P-graph associahedra, which are realized by repeated truncation of faces of simple polyhedra in accordance with tubes…
Deformation cones of graph associahedra and nestohedra
本文给出了图关联体和巢状体形变锥的刻面描述，推广了次模函数锥参数化，并构造了推广运动学关联体的运动学巢状体。
We prove a conjecture of Postnikov, Reiner and Williams by defining a partial order on the set of tree graphs with $n$ vertices that induces inequalities between the $γ$-polynomials of their associated graph-associahedra. The partial order is given by relating trees that can be obtained from one another by operations called tree shifts. We also show that tree shifts lower the $γ$-polynomials of graphs that are not trees, as do the flossing moves of Babson and Reiner.
The Malvenuto-Reutenauer algebra is a well-studied combinatorial Hopf algebra with a basis indexed by permutations. This algebra contains a wide variety of interesting sub Hopf algebras, in particular the Hopf algebra of plane binary trees introduced by Loday and Ronco. We compare two general constructions of subalgebras of the Malvenuto-Reutenauer algebra, both of which include the Loday-Ronco algebra. The first is a construction by Reading defined in terms of lattice quotients of the weak order, and the second is a construction by Ronco in terms of graph associahedra. To make this comparison…
A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation $m\circ n\equiv m+n-2$, the omitted lengths of colorful cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We prove that several but not all such monoids are realized. We then characterize exact Gallai graphs, i.…
In this paper, we prove that every 3-chromatic connected graph, except $C_7$, admits a 3-vertex coloring in which every vertex is the beginning of a 3-chromatic path. It is a special case of a conjecture due to S.~Akbari, F.~Khaghanpoor, and S.~Moazzeni, cited in [P.J. Cameron, Research problems from the BCC22, \emph{Discrete Math.} {\bf 311} (2011), 1074--1083], stating that every connected graph $G$ other than $C_7$ admits a $χ(G)$-coloring such that every vertex of $G$ is the beginning of a colorful path (i.e. a path of on $χ(G)$ vertices containing a vertex of each color). We also provide …
The diameter of type D associahedra and the non-leaving-face property
本文证明D型associahedron的图直径为2n-2，并证明所有BCD型associahedra满足非离开面性质，同时给出反例说明该性质并非普遍成立。
Given any finite graph, we offer a simple realization of the graph-associahedron polytope using integer coordinates.
Tropical varieties play an important role in algebraic geometry. The Bergman complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M generalize to matroids the notions of the tropical variety and positive tropical variety associated to a linear ideal. Our main result is that if A is a Coxeter arrangement of type Phi with corresponding oriented matroid M_Phi, then B+(M_Phi) is dual to the graph associahedron of type Phi, and B(M_Phi) equals the nested set complex of A. In addition, we prove that for any orientable matroid M, one can find |mu(M)| different reorientations of …
Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph $G$ encodes the combinatorics of search trees on $G$, defined recursively by a root $r$ together with search trees on each of the connected components of $G-r$. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. We give a tight bound of $Θ(m)$ on the diameter of trivially perfect graph associahedra on $m$ edge…
On the spectrum and expansion of graph associahedra
本文对图 associahedra 进行谱分析，给出其第二大特征值下界，利用公平划分分析 stellohedron 的谱，证明该多面体在特定区间存在特征值、有高重数特征值及两个小特征值。
Various results ensure the existence of large complete bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind, respectively due to Simonyi and Tardos (Combinatorica, 2006), Simonyi, Tardif, and Zsbán (The Electronic Journal of Combinatorics, 2013), and Chen (Journal of Combinatorial Theory, Series A, 2011). As a consequence of the generalization of Chen's theorem, we get new families of graphs whose chromatic number equals their circular chromatic number and that satisf…
A path in a vertex-colored graph is called a \emph{vertex-monochromatic path} if its internal vertices have the same color. A vertex-coloring of a graph is a \emph{monochromatic vertex-connection coloring} (\emph{MVC-coloring} for short), if there is a vertex-monochromatic path joining any two vertices in the graph. For a connected graph $G$, the \emph{monochromatic vertex-connection number}, denoted by $mvc(G)$, is defined to be the maximum number of colors used in an \emph{MVC-coloring} of $G$. These concepts of vertex-version are natural generalizations of the colorful monochromatic connect…
Using a $Z_q$-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a hypergraph).
A path in an edge-colored graph is called a \emph{monochromatic path} if all the edges on the path are colored the same. An edge-coloring of $G$ is a \emph{monochromatic connection coloring} (MC-coloring, for short) if there is a monochromatic path joining any two vertices in $G$. The \emph{monochromatic connection number}, denoted by $mc(G)$, is defined to be the maximum number of colors used in an MC-coloring of a graph $G$. These concepts were introduced by Caro and Yuster, and they got some nice results. In this paper, we will study two kinds of Erdős-Gallai-type problems for $mc(G)$, and …
M. Carr and S. Devadoss introduced in [7] the notion of tubing on a finite simple graph $Γ$, in the context of configuration spaces on the Hilbert plane. To any finite simple graph $Γ$ they associated a finite partially ordered set, whose elements are the tubings of $Γ$ and whose geometric realization is a convex polytope ${\mathcal K}Γ$, the graph-associahedron. For the complete graphs they recovered permutahedra, for linear graphs they got Stasheff's associahedra, while for simple graph they obtained the standard simplexes. The goal of the present work is to give an \emph{algebraic} descript…
It is known that a connected simple graph $G$ associates a simple polytope $P_G$ called a graph associahedron in Euclidean space. In this paper we show that the set of facet vectors of $P_G$ forms a root system if and only if $G$ is a cycle graph and that the root system is of type A.
An edge-coloring of the complete graph $K_n$ we call $F$-caring if it leaves no $F$-subgraph of $K_n$ monochromatic and at the same time every subset of $|V(F)|$ vertices contains in it at least one completely multicolored version of $F$. For the first two meaningful cases, when $F=K_{1,3}$ and $F=P_4$ we determine for infinitely many $n$ the minimum number of colors needed for an $F$-caring edge-coloring of $K_n$. An explicit family of $2\lceil\log_2 n\rceil$ $3$-edge-colorings of $K_n$ so that every quadruple of its vertices contains a totally multicolored $P_4$ in at least one of them is al…
Acyclic Poset Multiplihedra and their Quotients
研究利用图乘法多面体为无环关联多面体定义并实现新的乘法多面体，还对其商进行了研究，最终得出偏序集序多面体的区间多面体，为结合运算组合学提供了新的多面体相关内容。
Given a simple graph G, the graph associahedron KG is a simple polytope whose face poset is based on the connected subgraphs of G. This paper defines and constructs graph associahedra in a general context, for pseudographs with loops and multiple edges, which are also allowed to be disconnected. We then consider deformations of pseudograph associahedra as their underlying graphs are altered by edge contractions and edge deletions.
Maximal chains in lattices from graph associahedra: Tamari to the weak order
本文研究了推广弱序和Tamari格的极大管集格的极大链，主要针对在路图与完全图间插值的棒棒糖图，发现其格由满足312-避免推广的排列组成，并在Edelman-Greene双射下将最大长度链关联至部分可移表，进而求出相关对称函数的展开。
Almost-perfect colorful matchings in three-edge-colored bipartite graphs
该研究证明了三边着色二分图中存在满足各颜色边数指定条件的几乎完美彩色匹配，验证了Ryser-Brualdi-Stein猜想的三色多重扩展版本，且所得和的上界为最优。
We provide multicolored and infinite generalizations for a Ramsey-type problem raised by Bollobás, concerning colorings of $K_n$ where each color is well-represented. Let $χ$ be a coloring of the edges of a complete graph on $n$ vertices into $r$ colors. We call $χ$ $\varepsilon$-balanced if all color classes have $\varepsilon$ fraction of the edges. Fix some graph $H$, together with an $r$-coloring of its edges. Consider the smallest natural number $R_\varepsilon^r(H)$ such that for all $n\geq R_\varepsilon^r(H)$, all $\varepsilon$-balanced colorings $χ$ of $K_n$ contain a subgraph isomorphic…
Equivalence Between Four Models of Associahedra
本文研究了四类associahedra模型之间的等价性，揭示了Stasheff associahedra与Loday构造的同构关系，并探讨了multiplihedra和graph cubeahedra之间的联系。
The paper investigates connections between abstract polytopes and properly edge colored graphs. Given any finite n-edge-colored n-regular graph G, we associate to G a simple abstract polytope P_G of rank n, called the colorful polytope of G, with 1-skeleton isomorphic to G. We investigate the interplay between the geometric, combinatorial, or algebraic properties of the polytope P_G and the combinatorial or algebraic structure of the underlying graph G, focussing in particular on aspects of symmetry. Several such families of colorful polytopes are studied including examples derived from a Cayl…
An edge-coloring of a connected graph $G$ is called a {\it monochromatic connection coloring} (MC-coloring, for short), introduced by Caro and Yuster, if there is a monochromatic path joining any two vertices of the graph $G$. Let $mc(G)$ denote the maximum number of colors used in an MC-coloring of a graph $G$. Note that an MC-coloring does not exist if $G$ is not connected, and in this case we simply let $mc(G)=0$. We use $G(n,p)$ to denote the Erdös-Rényi random graph model, in which each of the $\binom{n}{2}$ pairs of vertices appears as an edge with probability $p$ independently from othe…
Given a graph G, we construct a simple, convex polytope whose face poset is based on the connected subgraphs of G. This provides a natural generalization of the Stasheff associahedron and the Bott-Taubes cyclohedron. Moreover, we show that for any simplicial Coxeter system, the minimal blow-ups of its associated Coxeter complex has a tiling by graph-associahedra. The geometric and combinatorial properties of the complex as well as of the polyhedra are given. These spaces are natural generalizations of the Deligne-Knudsen-Mumford compactification of the real moduli space of curves.
The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f-, h- and gamma-vectors. These polytopes include permutohedra, associahedra, graph-associahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas for h-vectors and gamma-vectors involving descent statistics. This includes a combinatorial interpretation for gamma-vectors of a large class of generalized permutohedra which are flag simple polytopes, and confirms for them Gal's conjecture on nonnegativity of gamma-vectors. We calculate expli…
Combining Ky Fan's theorem with ideas of Greene and Matousek we prove a generalization of Dol'nikov's theorem. Using another variant of the Borsuk-Ulam theorem due to Bacon and Tucker, we also prove the presence of all possible completely multicolored t-vertex complete bipartite graphs in t-colored t-chromatic Kneser graphs and in several of their relatives. In particular, this implies a generalization of a recent result of G. Spencer and F. E. Su.
Effect of graph operations on graph associahedra
本研究探讨添加单纯顶点、真/假孪生顶点、删除真孪生诱导子图边集等图运算对旋转图结构的影响，推导色数保持条件，给出直径新下界并算出部分完全二部图旋转图直径精确值。
The caterpillar associahedron $\mathcal{A}(G)$ is a polytope arising from the rotation graph of search trees on a caterpillar tree $G$, generalizing the rotation graph of binary search trees (BSTs) and thus the conventional associahedron. We show that the diameter of $\mathcal{A}(G)$ is $Θ(n + m \cdot (H+1))$, where $n$ is the number of vertices, $m$ is the number of leaves, and $H$ is the entropy of the leaf distribution of $G$. Our proofs reveal a strong connection between caterpillar associahedra and searching in BSTs. We prove the lower bound using Wilber's first lower bound for dynamic BS…
The associahedron is a convex polytope whose face poset is based on nonintersecting diagonals of a convex polygon. In this paper, given an arbitrary simple polygon P, we construct a polytopal complex analogous to the associahedron based on convex diagonalizations of P. We describe topological properties of this complex and provide realizations based on secondary polytopes. Moreover, using the visibility graph of P, a deformation space of polygons is created which encapsulates substructures of the associahedron.
To any graph $G$ one can associate a toric variety $X(\mathcal{P}G)$, obtained as a blowup of projective space along coordinate subspaces corresponding to connected subgraphs of $G$. The polytope of this toric variety is the graph associahedron of $G$, a class of polytopes that includes the permutohedron, associahedron, and stellahedron. We show that the space $X(\mathcal{P}{G})$ is isomorphic to a Hassett compactification of $M_{0,n}$ precisely when $G$ is an iterated cone over a discrete set. This may be viewed as a generalization of the well-known fact that the Losev--Manin moduli space is …
Geometric realizations of $ν$-associahedra via brick polyhedra
本文利用砖多面体实现了$\ u$-associahedron的几何构造，其边图为$\ u$-Tamari格，并给出了优雅的顶点坐标描述。
Colorful Hamilton cycles in random graphs
研究随机图$G_{n,p}$中边随机着色时哈密顿圈的颜色分布轮廓，确定其达到全向量集合的阈值。
The Erdős-Pósa Property for Colorful Minors
该研究确定了具有彩色子式关系的Erdős-Pósa性质的彩色图，通过结构形式、障碍集、网格形式三种等价形式完成刻画，拓展了经典的Erdős-Pósa性质理论。
From the paper of the first author it follows that upper and lower bounds for $γ$-vector of a simple polytope imply the bounds for its $g$-,$h$- and $f$-vectors. In the paper of the second author it was obtained unimprovable upper and lower bounds for $γ$-vectors of flag nestohedra, particularly Gal's conjecture was proved for this case. In the present paper we obtain unimprovable upper and lower bounds for $γ$-vectors (consequently, for $g$-,$h$- and $f$-vectors) of graph-associahedra and some its important subclasses. We use the constructions that for an $(n-1)$-dimensional graph-associahedr…
Colorful Minors
本文提出彩色极小图概念，发展其结构理论，并展示其在算法中的应用。
We consider a natural notion of search trees on graphs, which we show is ubiquitous in various areas of discrete mathematics and computer science. Search trees on graphs can be modified by local operations called rotations, which generalize rotations in binary search trees. The rotation graph of search trees on a graph $G$ is the skeleton of a polytope called the graph associahedron of $G$. We consider the case where the graph $G$ is a tree. We construct a family of trees $G$ on $n$ vertices and pairs of search trees on $G$ such that the minimum number of rotations required to transform one se…
We investigate a family of polytopes introduced by E.M.\ Feichtner, A.\ Postnikov and B.\ Sturmfels, which were named nestohedra. The vertices of these polytopes may intuitively be understood as constructions of hypergraphs. Limit cases in this family of polytopes are, on the one end, simplices, and, on the other end, permutohedra. In between, as notable members one finds associahedra and cyclohedra. The polytopes in this family are investigated here both as abstract polytopes and as realized in Euclidean spaces of all finite dimensions. The later realizations are inspired by J.D.\ Stasheff's …
$P$-associahedra
该研究针对偏序集$P$构造了$P$-结合多面体$A(P)$，引入仿射偏序环面多面体并将其实现为圆周上$n$点构型空间的紧化，可得到多种经典多面体作为特例。
On a rainbow extremal problem for color-critical graphs
本文研究颜色临界图的彩虹极值问题，证明 Keevash 等人关于 4-颜色临界图及几乎所有 r>4 颜色临界图的猜想，并说明非颜色临界非二分图的两种自然构造均非极值。
Computing distances is FPT on graph associahedra and W[2]-hard on hypergraphic polytopes
研究在图关联多面体上计算距离的FPT算法，并证明在超图多面体上该问题的W[2]-难性，同时探讨其近似难性。
Hassler Whitney's theorem of 1931 reduces the task of finding proper, vertex 4-colorings of triangulations of the 2-sphere to finding such colorings for the class \(\mathfrak H\) of triangulations of the 2-sphere that have a Hamiltonian circuit. This has been used by Whitney and others from 1936 to the present to find equivalent reformulations of the 4 Color Theorem (4CT). Recently there has been activity to try to use some of these reformuations to find a shorter proof of the 4CT. Every triangulation in \(\mathfrak H\) has a dual graph that is a union of two binary trees with the same number …
Listing faces of polytopes
本文提出了一种高效算法，用于构造多面体的面哈密顿回路，解决了B-排列多面体的面哈密顿回路猜想。
Compatibility fans for graphical nested complexes
本文受有限型簇复形类比启发，将Fomin-Zelevinsky相容性扇构造推广至图形嵌套复形，定义管间相容度，证明其相容向量支撑完整单纯扇，实现图嵌套复形，并恢复路径情形的加泰罗尼亚多个结合多面体实现。
Facial nested complexes and acyclonestohedra
本文研究有向拟阵面格上嵌套复形的多面体实现，通过将面嵌套复形嵌入布尔嵌套复形并构造无环嵌套多面体，统一推广了偏序集关联多面体等多类嵌套多面体族。
Small covers of graph-associahedra and realization of cycles
本文研究图结合多面体小覆盖的普适实现性质，证明连通图对应的小覆盖或其二重定向覆盖为 URC-流形，并给出 Stasheff 结合多面体的更简单例子。
Random walks on bounded degree expander graphs have numerous applications, both in theoretical and practical computational problems. A key property of these walks is that they converge rapidly to their stationary distribution. In this work we {\em define high order random walks}: These are generalizations of random walks on graphs to high dimensional simplicial complexes, which are the high dimensional analogues of graphs. A simplicial complex of dimension $d$ has vertices, edges, triangles, pyramids, up to $d$-dimensional cells. For any $0 \leq i < d$, a high order random walk on dimension $i…
Hardness of Finding Combinatorial Shortest Paths on Graph Associahedra
该研究证明图associahedron顶点间组合最短路径求解为NP难，解决了相关开放问题，并推论多拟阵基多胞体的该类路径无法多项式时间求解，揭示拟阵与多拟阵的计算差异。
An associahedron is a polytope whose vertices correspond to the triangulations of a convex polygon and whose edges correspond to flips between them. A particularly elegant realization of the associahedron, due to S. Shnider and S. Sternberg and popularized by J.-L. Loday, has been generalized in two directions: on the one hand by A. Postnikov to obtain a realization of the graph associahedra of M. Carr and S. Devadoss, and on the other hand by C. Hohlweg and C. Lange to obtain multiple realizations of the associahedron parametrized by a sequence of signs. The goal of this paper is to unify and…
Motivated by the graph associahedron KG, a polytope whose face poset is based on connected subgraphs of G, we consider the notion of associativity and tubes on posets. This leads to a new family of simple convex polytopes obtained by iterated truncations. These generalize graph associahedra and nestohedra, even encompassing notions of nestings on CW-complexes. However, these poset associahedra fall in a different category altogether than generalized permutohedra.
Differential algebra of polytopes and inversion formulas
本文利用多胞体的微分代数阐明结合多胞体和置换多胞体与形式幂级数反演公式的深刻联系，并探讨其与Deligne-Mumford模空间及Faà di Bruno公式的关联。
Let G=(V,A) be a vertex-colored arc-weighted directed acyclic graph (DAG) rooted in some vertex r, and let H be its color hierarchy graph, defined as follows: V(H) is the color set C of G, and an arc from color c to color c' exists in H if there is an arc in G from a vertex of color c to a vertex of color c'. In this paper, we study the MAXIMUM COLORFUL ARBORESCENCE problem (or MCA), which takes as input a DAG G with the additional constraint that H is also a DAG, and aims at finding in G an arborescence rooted in r, of maximum weight, and in which no color appears more than once. The MCA prob…
Numerous applications in scheduling, such as resource allocation or steel manufacturing, can be modeled using the NP-hard Independent Set problem (given an undirected graph and an integer k, find a set of at least k pairwise non-adjacent vertices). Here, one encounters special graph classes like 2-union graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise unions of an interval graph and a cluster graph), on which Independent Set remains NP-hard but admits constant-ratio approximations in polynomial time. We study the parameterized complexity of Independent Set on 2-unio…
Nestohedra are a family of convex polytopes that includes permutohedra, associahedra, and graph associahedra. In this paper, we study an extension of such polytopes, called extended nestohedra. We show that these objects are indeed the boundaries of simple polytopes, answering a question of Lam and Pylyavskyy. We also study the duals of (extended) nestohedra, giving a complete characterization of isomorphisms (as simplicial complexes) between the duals of extended nestohedra and a partial characterization of isomorphisms between the duals of nestohedra and extended nestohedra. In addition, we …
Reconnectads
本文引入reconnectad这一类operad新结构，以有限简单图为输入、通过子图重连补集实现合成，以Carr和Devadoss图结合体复曲面簇为典型例子，发展其一般理论并用于研究由同调群组装的“奇妙reconnectad”。
Rainbow Separating Path Systems
本文提出一种彩色分离路径系统，研究不同图类和颜色数下的最小系统规模，并发现随着颜色数增加，某些图类呈现三种渐进行为。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。