A geometric approach to acyclic orientations
Richard Ehrenborg, MLE Slone
具有给定sink的连通图的无圈方向集具有自然偏序集结构。本文给出了Jim Propp的一个结果的几何证明：这个偏序集是分配格的不交并。
The set of acyclic orientations of a connected graph with a given sink has a natural poset structure. We give a geometric proof of a result of Jim Propp: this poset is the disjoint union of distributive lattices.
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We define the acyclic orientation polynomial of a graph to be the generating function for the sinks of its acyclic orientations. Stanley proved that the number of acyclic orientations is equal to the chromatic polynomial evaluated at $-1$ up to sign. Motivated by this link between acyclic orientations and the chromatic polynomial, we develop "acyclic orientation" analogues of theorems concerning the chromatic polynomial of Birkhoff, Whitney, and Greene-Zaslavsky. As an application, we provide a new proof for Stanley's sink theorem for chromatic symmetric functions $X_G$. This theorem gives a r…
Sinks in Acyclic Orientations of Graphs
Greene和Zaslavsky证明了具有唯一sink的图的无圈定向数等于其色多项式的线性系数。我们给出了三个新的证明，这一结果使用纯感应，非交换对称函数，和算法双射。
Encoding and Enumerating Acyclic Orientations of Graphs
本文研究图的非循环定向编码与枚举，提出完全多部图的向量编码和递归构造方法，并给出若干闭公式及任意图非循环定向数目的精确上下界。
The orientation completion problem for a fixed class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class. Orientation completion problems have been studied recently for several classes of oriented graphs, yielding both polynomial time solutions and NP-completeness results. Local tournaments are a well-structured class of oriented graphs that generalize tournaments and their underlying graphs are intimately related to proper circular-arc graphs. Proper interval graphs are precisely those which can be oriented as acyclic local tourn…
Acyclic orientations and Hessenberg varieties
本文建立 Dyck 图无环定向与正则幂零 Hessenberg 簇 Tymoczko 胞腔的双射，并由此推出 Shareshian-Wachs 公式。
The main contribution of this paper is a formula for the number of acyclic orientations of a complete bipartite, $K_{n_1,n_2},$ revealing that it is equal to the poly-Bernoulli number $B_{n_1}^{(-n_2)}$ introduced in 1997 by Kaneko. We also give a simple bijective identification of acyclic orientations and lonesum matrices, confirming a 2008 result of Brewbaker, and show that the poly-Bernoulli numbers behave concavely, as well being symmetric. A second goal is to explore the behaviour of more general complete $r$-partite graphs in the space of the number of acyclic orientations. We prove that…
The Tutte Polynomial of a Morphism of Matroids 6. A Multi-Faceted Counting Formula for Hyperplane Regions and Acyclic Orientations
证明了有序定向拟阵的某些定向相关参数的4元母函数是其Tutte多项式在（x + u，y+v）处的求值.这种评价包含作为特殊情况下的超平面安排和非循环的方向图中的区域计数。几个新的2-变量展开的Tutte多项式的定向拟阵作为推论。这一结果持有更普遍的定向拟阵的角度来看，与特定的特殊情况下，计数的有界区域超平面安排或双极非循环的方向图。在推论中，我们得到了Tutte多项式的偏导数作为相同取向参数的生成函数的表达式。
Hamiltonicity of graphs of acyclic orientations and acyclic polynomials
该研究刻画了$\text{AO}$-哈密顿的多路径，提出保持$\text{AO}$-哈密顿性的粘合准则与方法，定义无环多项式并探究其性质，为$\text{AO}$-哈密顿性的归纳认证提供支撑。
Building on previous work by Cameron et al. in [3], we give a recurrence for computing the number of acyclic orientations of complete $k$-partite graphs, which can be implemented to obtain a dynamic programming algorithm running in time $n^{O(k)}$, where $n$ is the number of vertices in the graph. We prove our result by using a relationship between the number of acyclic orientations and the number of Hamiltonian paths in complete $k$-partite graphs and providing a recurrence for the latter quantity. We give a simple extension of our algorithm to the situation when we are an edge removal away f…
We present a new correspondence between acyclic orientations and coloring of a signed graph (symmetric graph). Goodall et al. introduced a bivariate chromatic polynomial $χ_G(k,l)$ that counts the number of signed colorings using colors $0,\pm1,\dots,\pm k$ along with $l-1$ symmetric colors $0_1,\dots,0_{l-1}$. We show that the evaluation of the bivariate chromatic polynomial $|χ_G(-1,2)|$ is equal to the number of acyclic orientations of the signed graph modulo the equivalence relation generated by swapping sources and sinks. We present three proofs of this fact, a proof using toric hyperplan…
Acyclic Orientations and Spanning Trees
引入了与单图的部分无圈定向相关联的多面体胞腔复形，推广了无圈定向。利用细胞分辨率的理论，观察到这些多面体胞腔复形的两个最低限度地解决某些特殊的组合多项式理想有关的非循环方向。这些理想被明确地发现是亚历山大对偶，这相对于文献中的可比结果，概括在一个更干净，更有启发性的方式permutohedron和树理想之间的著名的对偶。这些结果的组合学基础自然导致一个典型的方式来表示一个标记的单图的根生成森林的非交叉树，这些表示被观察到携带大量的信息广义树理想和非循环方向的图，和非交叉分区的全序集。收集和组织这些信息的枚举和结构性后果的一个小样本进行了详细研究。然后介绍和探讨了这一组合理论的应用，即：状态空间上的随机过程等于一个单图的所有非循环方向的集合，包括不可约的马尔可夫链，它表现出从基于线性扩张到均匀的平稳分布;和一个纯粹的代数介绍的主要问题，在引导渗透，可能使它易于探索的一个图的集合的所有扩展集与计算机。
We extend the duality between acyclic orientations and totally cyclic orientations on planar graphs to dualities on graphs on orientable surfaces by introducing boundary acyclic orientations and totally bi-walkable orientations. In addition, we provide a reciprocity theorem connecting local tensions and boundary acyclic orientations. Furthermore, we define the balanced flow polynomial which is connected with tension polynomial by duality and with totally bi-walkable orientations by reciprocity.
The chromatic polynomial $P(G,x)$ of a graph $G$ of order $n$ can be expressed as $\sum\limits_{i=1}^n(-1)^{n-i}a_{i}x^i$, where $a_i$ is interpreted as the number of broken-cycle free spanning subgraphs of $G$ with exactly $i$ components. The parameter $ε(G)=\sum\limits_{i=1}^n (n-i)a_i/\sum\limits_{i=1}^n a_i$ is the mean size of a broken-cycle-free spanning subgraph of $G$. In this article, we confirm and strengthen a conjecture proposed by Lundow and Markström in 2006 that $ε(T_n)< ε(G)<ε(K_n)$ holds for any connected graph $G$ of order $n$ which is neither the complete graph $K_n$ nor a t…
We study graphs which admit an acyclic orientation that contains an out-branching and in-branching which are arc-disjoint (such an orientation is called {\bf good}). A {\bf 2T-graph} is a graph whose edge set can be decomposed into two edge-disjoint spanning trees. Clearly a graph has a good orientation if and only if it contains a spanning 2T-graph with a good orientation, implying that 2T-graphs play a central role. Vertex-minimal 2T-graphs with at least two vertices, also known as {\bf generic circuits}, play an important role in rigidity theory for graphs. It was shown in \cite{bangGOpaper…
In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. Firstly, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage-Squire-West construction with a recent algorithm for generating elimination trees of chordal graphs. Secondly, we consider quotients of lattices of acyclic orientations of ch…
Acyclic orientations of mixed graphs
研究无环混合图的定向完成问题，证明若干判定问题的复杂性，并给出一个多项式算法，提出开放问题。
In 1995, Richard Stanley introduced the chromatic symmetric function $X_G$ of a graph $G$ and proved that, when written in terms of the elementary symmetric functions, it reveals the number of acyclic orientations of $G$ with a given number of sinks. In this paper, we generalize this result to signed graphs, that is, to graphs whose edges are labeled with $+$ or $-$ and whose colorings and orientations can interact with their signs. Additionally, we introduce a non-homogeneous basis which detects the number of sinks and which not only gives a Stanley-type result for signed graphs but gives an …
Note about the complexity of the acyclic orientation with parity constraint problem
本文研究了具有奇偶约束的无环定向问题的复杂性，证明了在某些图中该问题为NP难。
$G$-Parking Functions, Acyclic Orientations and Spanning Trees
给定一个无向图$G=(V,E)$和一个指定的顶点$q\in V$，一个$G$ -停车函数（相对于$q$）的概念是独立开发和研究的各种作者，最近获得了新的关注。这个概念推广了与完全图相关的停车函数的经典概念。本文研究了{ \em极大} $G$ -停车函数的性质，并给出了它们与无断路的$G$的生成树集之间的一个新的双射.作为一个案例研究，我们专门我们的一些结果，对应于离散$n$立方体$Q_n$的图形。我们目前的文章在一个暂时的独立的形式，因为我们发现组合方面的$G$停车功能有点分散在文献中，通常与沙堆模型和密切相关的芯片发射游戏一起处理。
Some polynomial classes for the acyclic orientation with parity constraint problem
本文研究了无环定向问题中的奇偶约束，提出多项式类以确定无环T-奇定向的存在条件，并通过构造性证明展示了其在特定图结构中的可解性。
Full Orientability of the Square of a Cycle
设D是单图G的无圈定向。如果D的弧的反转产生一个有向环，则称之为从属弧。设d（D）表示D中相关弧的个数。定义m和M为G的所有非循环方向D上d（D）的最小和最大个数。我们称G是完全可定向的，如果G有一个非循环定向，其中对任意k满足m <= k <= M，恰好有k条依赖弧.本文证明了长为n的圈C_n的平方除n=6外都是完全可定向的。
Recall that an excedance of a permutation $π$ is any position $i$ such that $π_i > i$. Inspired by the work of Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it gets sorted by a certain sequence of toppling moves. One of our main results is that the number of toppleable permutations on $n$ letters is the same as those for which excedances happen exactly at $\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Additionally, we show that the above is also the number of acyclic orientations with unique sink (AUSOs) of the complete bipa…
Given an acyclic oriented graph $\vec{H}$ and a graph $G$, we write $G \to \vec{H}$ if every orientation of $G$ has an oriented copy of $\vec{H}$. We define $\vec{R}(\vec{H})$ as the smallest number $n$ such that there exists a graph $G$ satisfying $G \to \vec{H}$. Denoting by $R(H)$ the classical Ramsey number of a graph $H$, we show that $\vec{R}(\vec{H}) \leq 2R(H)^{c \log^2 h}$ for every acyclic oriented graph $\vec{H}$ with $h$ vertices, where $H$ is its underlying undirected graph. We also study the threshold function for the event $\{G(n,p) \to \vec{H}\}$ in the binomial random graph $G…
Finding an Unknown Acyclic Orientation of a Given Graph
设c（G）是为了确定给定图G在最坏情况下的未知非循环方向而必须测试的最小边数。例如，如果G是n个顶点上的完全图，则c（G）是对n个数进行排序所需的最小比较次数。本文证明了对任意n阶图G，c（G）\le（1/4+o（1））n^2，从而肯定地回答了Aigner，Triesch和Tuza [Discrete Mathematics，144（1995）3-10]提出的一个问题.此外，我们还证明了，对于每个e>0，在乘法因子74/73-e内逼近参数c（G）是NP-困难的。
The distribution of the length of the longest path in random acyclic orientations of a complete bipartite graph
本文研究了完全二部图随机无环定向中最长路径长度的分布，给出了概率生成函数，并利用解析组合学证明了在两部分大小相等趋于无穷时分布渐近服从高斯分布，回答了 Peter J. Cameron 的问题。
The orientation completion problem for a class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class by orienting the unoriented edges of the partially oriented graph. Orientation completion problems have been studied recently for several classes of oriented graphs, yielding both polynomial time solutions as well as NP-completeness results. Local tournaments are a well-structured class of oriented graphs that generalize tournaments and their underlying graphs are intimately related to proper circular-arc graphs. According to Skrien,…
In this note we study the complexity of some generalizations of the notion of $st$-numbering. Suppose that given some functions $f$ and $g$, we want to order the vertices of a graph such that every vertex $v$ is preceded by at least $f(v)$ of its neighbors and succeeded by at least $g(v)$ of its neighbors. We prove that this problem is solvable in polynomial time if $fg\equiv 0$, but it becomes NP-complete for $f\equiv g \equiv 2$. This answers a question of the first author posed in 2009.
Sortable Elements for Quivers with Cycles
本文将Coxeter群中c-排序元素的概念推广到任意定向的箭图Ω，定义了Ω-排序元素，并证明了其关键性质（如通过约化到无环情形），为推广到非无环簇代数奠定基础。
Kalai orientations on matroid polytopes
设P是一个多胞形，G（P）是P的一个图.根据Gil Kalai，我们说G（P）的一个无圈定向O是好的，如果对P的每一个非空面F，导出图G（F）恰好有一个sink.吉尔·卡莱给出了一个简单的方法来区分一个简单的多面体和它的图的好方向。这篇文章是一个更广泛的研究“好的方向”（图）上拟阵多面体。
In an earlier paper, the first two authors defined orientations on hypergraphs. Using this definition we provide an explicit bijection between acyclic orientations in hypergraphs and faces of hypergraphic polytopes. This allows us to obtain a geometric interpretation of the coefficients of the antipode map in a Hopf algebra of hypergraphs. This interpretation differs from similar ones for a different Hopf structure on hypergraphs provided recently by Aguiar and Ardila. Furthermore, making use of the tools and definitions developed here regarding orientations of hypergraphs we provide a charact…
Equivalences on Acyclic Orientations
循环群和二面体群可以作用于无向图Y的无循环定向集Acyc（Y），这分别产生等价关系~kappa和~delta.这两个行动和他们相应的等价类是密切相关的组合问题的背景下出现的考克斯特群，顺序动力系统，芯片发射游戏，并表示颤抖。本文构造了具有顶点集Acyc（Y）的图C（Y）和D（Y），其连通分支表示等价类。这些图的连通分量的数量分别表示为kappa（Y）和delta（Y）。我们刻画了C（Y）和D（Y）的结构，展示了如何从kappa（Y）导出delta（Y），并给出了kappa（Y）的计数结果.此外，我们展示了如何将偏序集结构与每个kappa-等价类相关联，并刻画了这些偏序集。这允许我们创建从Acyc（Y）/~kappa到Acyc（Y '）/~kappa和Acyc（Y“）/~kappa的并集的双射，Y'和Y”分别表示Y中的循环边的边删除和边收缩，这反过来表明kappa（Y）可以通过在（1，0）处对Tutte多项式求值来获得。
Let $G$ be a simple graph and let $\mathcal{L}(G)$ be the free partially commutative Lie algebra associated to $G$. In this paper, using heaps of pieces, we prove an expression for the generalized $\textbf k$-chromatic polynomial of $G$ in terms of dimensions of the grade spaces of $\mathcal{L}(G)$. This will give us a new interpretation for the chromatic polynomials in terms of multilinear heaps and Lyndon length. The classical results of Stanley, and Greene and Zaslavsky regarding the acyclic orientations of $G$ are obtained as corollaries. A heap with a unique minimal piece is said to be a …
Chordal Graphs are Fully Orientable
设D是图G的无圈定向。如果D的弧的反转产生一个有向环，则称之为从属弧。设m和M表示G的所有非循环定向上的依赖弧数的最小值和最大值。我们称G是完全可定向的，如果G有一个非循环定向，其中对任意d满足m <= d <= M，恰有d条依赖弧.一个图G称为弦图，如果G中每个长度至少为4的圈都有一个弦。我们证明了所有弦图都是完全可定向的。
Given an undirected graph $G = (V, E)$, and a vertex $r\in V$, an $r$-acyclic orientation of $G$ is an orientation $OE$ of the edges of $G$ such that the digraph $OG = (V, OE)$ is acyclic and $r$ is the unique vertex with indegree equal to 0. For $w\in \mathbb{R}^E_+$, $k(G, w)$ is the value of the $w$-maximum packing of $r$-arborescences for all $r\in V$ and all $r$-acyclic orientations $OE$ of $G$. In this case, the Broadcast Routing (in Computers Networks) Problem (BRP) is to compute $k(G, w)$, by finding an optimal $r$ and an optimal $r$-acyclic orientation. BRP is a mathematical formulati…
Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs
\emph{Golomb ruler}是不同整数的序列（统治者的\emph{markings}），其成对差异是不同的。Golomb规则可以追溯到20世纪30年代的加法数论，并吸引了最近对存在问题的研究活动，例如寻找\emph{optimal} Golomb规则（给定固定数量的标记的最小长度）。我们的目标是以系统的方式枚举Golomb规则：我们研究[g_m（t）：= # { \x \in \Z^{m+1}：\，0 = x_0 < x_1 <. < x_{m-1} < x_m = t，\text{all} x_j - x_k \text{distinct} }，]具有$m+1$标记和长度$t$的哥伦布标尺的数量。我们的主要结果是，$g_m(t)$是一个准多项式在$t$满足组合互易定理：$(-1)^{m-1} g_m(-t)$等于统治者的数量$\x$的长度$t$与$m+1$标记，每个计数与其\emph{Golomb multiplicity}，这措施有多少组合不同的哥伦布统治者是在一个小邻域的$\x$。我们的互易定理可以用与Golomb规则相关的某些混合图来解释;在这种语言中，它让人想起Stanley的色多项式互易定理。因此，在本文的第二部分中，我们开发了一个类似的斯坦利定理的混合图，连接他们的色多项式的非循环方向。
We study the size of certain acyclic domains that arise from geometric and combinatorial constructions. These acyclic domains consist of all permutations visited by commuting equivalence classes of maximal reduced decompositions if we consider the symmetric group and, more generally, of all c-singletons of a Cambrian lattice associated to the weak order of a finite Coxeter group. For this reason, we call these sets Cambrian acyclic domains. Extending a closed formula of Galambos--Reiner for a particular acyclic domain called Fishburn's alternating scheme, we provide explicit formulae for the s…
Enumeration of paths and cycles and e-coefficients of incomparability graphs
证明了无圈有向图的补图上的Hamilton路数等于圈覆盖数。作为应用，我们得到了不可比图的色对称函数用初等对称函数的一个新的展开式。分析一些组合的影响，这种扩张导致三个双射涉及非循环方向。
If $G$ is a graph and $\vec H$ is an oriented graph, we write $G\to \vec H$ to say that every orientation of the edges of $G$ contains $\vec H$ as a subdigraph. We consider the case in which $G=G(n,p)$, the binomial random graph. We determine the threshold $p_{\vec H}=p_{\vec H}(n)$ for the property $G(n,p)\to \vec H$ for the cases in which $\vec H$ is an acyclic orientation of a complete graph or of a cycle.
Arc weighted acyclic orientations and variations of degeneracy of graphs
本文推广无环定向至弧赋权定向，定义新的$ST^{(2)}$-$f$-退化度变体，否定Richter关于3-连通非完全平面图截断度可选性的问题，证明此类图及真子图闭族中对应图的相关可选性。
Shift Graphs, Chromatic Number and Acyclic One-Path Orientations
本文利用移位图及其诱导子图，改进了完全二分图情形下无F诱导子图的色数上界，并证明移位图无AOP性质，同时构造了高奇围长但无AOP性质的图。
For an oriented graph $G$, let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of $G$. Let $f(n)$ be the smallest integer such that every oriented graph $G$ with chromatic number larger than $f(n)$ has $f(G) > n$. Let $g(n)$ be the smallest integer such that every tournament $G$ with more than $g(n)$ vertices has $f(G) > n$. It is straightforward that $Ω(n) \le g(n) \le f(n) \le n^2$. This paper provides the first nontrivial lower and upper bounds for $g(n)$. In particular, it is proved that $\frac{1}{4}n^{8/7} \le g(n) \le n^2-(2-\frac{1}{\sqrt{2}})n+2$. It is also shown tha…
The Merino-Welsh Conjecture holds for Series-Parallel Graphs
Merino-Welsh猜想认为，图的生成树的个数不大于该图的全循环定向和非循环定向的个数中的最大值.我们证明了这个猜想的系列平行图类。
e-basis Coefficients of Chromatic Symmetric Functions
本文推广 Stanley 关于色对称函数 e-基系数与无环定向汇点序列的结果，证明顶点加权版本，并提出可解释更多 e-系数和的更强猜想。
Orientations, lattice polytopes, and group arrangements I: Chromatic and tension polynomials of graphs
这是第一个一系列的文件协会的方向，格多面体，和阿贝尔群安排图。目的是解释图在零和负整数上的整张力多项式和模张力多项式。整个论述是在子群排列和Ehrhart多项式的应用的框架下进行的。这种观点导致以下主要结果的文件：（i）的互易性法律的整体张力多项式;（ii）的互易性法律的模张力多项式;和（iii）的Tutte多项式$T(G;x,y)$的值的一个新的解释图$G$在$(1,0)$的非循环方向的切割等价类的数量$G$。
The Minimum Number of Dependent Arcs and a Related Parameter of Generalized Mycielski Graphs
设D是图G的非圈定向。如果D的弧的反转创建有向循环，则D的弧是从属的。设m（G）表示G的所有非循环定向上的最小相依弧数.对任意k > 0，定义了G的广义Mycielski图M_k（G）.注意M_1（G）是G的通常Mycielskian。推广了关于K中m（M_1（G））的结果. L. Collins，K. Tysdal，J.图论，46（2004），285-296，to m（M_k（G））. Hasse图的底层图称为覆盖图。设c（G）表示图G中为得到覆盖图所需删除的最少边数。同时也得到了关于c（G）的一些结果.
Separable convex optimization over indegree polytopes
本文研究无向图在入度目标下的无环定向优化，证明多种公平性目标（如凸函数和、递减最小化等）的NP难度，同时给出加权最小最后排序等多项式算法及近似算法，刻画了算法边界并应用于调度与无死锁路由。
Acyclic sets and colorings in digraphs under restrictions on degrees and cycle lengths
本文研究了有向图中无环集和着色问题在度数和环长限制下的性质，证明了在随机有向图中无环集大小的渐进行为，并提出了相关猜想。
We develop a new framework for investigating linear equivalence of divisors on graphs using a generalization of Gioan's cycle--cocycle reversal system for partial orientations. An oriented version of Dhar's burning algorithm is introduced and employed in the study of acyclicity for partial orientations. We then show that the Baker--Norine rank of a partially orientable divisor is one less than the minimum number of directed paths which need to be reversed in the generalized cycle--cocycle reversal system to produce an acyclic partial orientation. These results are applied in providing new proo…
Partial Graph Orientations and the Tutte Polynomial
Gessel和Sagan使用深度优先搜索研究了Tutte多项式$T(x,y)$，并应用他们的技术证明了图的非循环部分方向的数量是$2^gT(3,1/2)$。我们提供了一个简短的删除压缩证明这个结果，并证明对偶，强连接的部分方向的数量是$2^{n-1}T(1/2,3)$。证明了部分方向模循环反转的个数为$2^gT(3,1)$，部分方向模割反转的个数为$2^{n-1}T(1,3)$ .为了证明这些结果，我们引入切割和循环最小的部分方向，提供了杰出的代表部分方向模切割和循环反转。这些扩展类的总方向介绍了Gioan，格林和Zaslavksy，我们强调了密切的联系与图形和cographic劳伦斯理想。我们的结论与边色概括的数量，这使得一个新的解释的可靠性多项式的所有概率，$p$与$0 < p <1/2$。
On the number and size of Markov equivalence classes of random directed acyclic graphs
本文研究了均匀随机有向无环图的 Markov 等价类，利用塔分解证明了等价类与图的比值趋于正常数，且典型等价类大小有界并具有超多项式尾部。
Geometric bijections between spanning subgraphs and orientations of a graph
该研究将Backman等人提出的图生成树与定向间的几何双射拓展至子图-定向对应，给出纯组合证明及相关几何证明，且结论适用于正则拟阵。
Chain--collider--fork Decompositions of Transitive Tournament
研究传递锦标赛中链、碰撞和叉三种两弧基序的分解与打包，给出最大弧不交拷贝数。
We study the map from conductances to edge energies for harmonic functions on finite graphs with Dirichlet boundary conditions. We prove that for any compatible acyclic orientation and choice of energies there is a unique choice of conductances such that the associated harmonic function realizes those orientations and energies. We call the associated function enharmonic. For rational energies and boundary data the Galois group of ${\mathbb Q}^{tr}$ (the totally real algebraic numbers) over ${\mathbb Q}$ permutes the enharmonic functions, acting on the set of compatible acyclic orientations. A …
Shortest paths on polymatroids and hypergraphic polytopes
该研究针对广义排列多面体顶点间的最短路径问题，证明其在简单多拟阵、超图多胞体等场景下的NP难与APX难性质，同时给出超图无环定向翻转距离的近似算法及线性超图下的精确多项式算法。
Given an undirected graph, one can assign directions to each of the edges of the graph, thus orienting the graph. To be as egalitarian as possible, one may wish to find an orientation such that no vertex is unfairly hit with too many arcs directed into it. We discuss how this objective arises in problems resulting from telecommunications. We give optimal, polynomial-time algorithms for: finding an orientation that minimizes the lexicographic order of the indegrees and finding a strongly-connected orientation that minimizes the maximum indegree. We show that minimizing the lexicographic order o…
Combinatorial Properties for a Class of Simplicial Complexes Extended from Pseudo-fractal Scale-free Web
本文研究伪分形无标度网络扩展的一类单纯复形的组合性质，显式确定了独立数、支配数和色数，并推导了无环定向、生成树及完美匹配等数量的闭式表达式。
On Phases of Unique Sink Orientations
本文研究 USO 中相位的结构，给出 O(3^n) 的相位计算算法，并证明电路简洁编码的无环 USO 中边同相位判定问题是 PSPACE 完全的。
Stanley in his paper [Stanley, Richard P.: Acyclic orientations of graphs In: Discrete Mathematics 5 (1973), Nr. 2, S. 171-178.] provided interpretations of the chromatic polynomial when it is substituted with negative integers. Greene and Zaslavsky interpreted the coefficients of the chromatic polynomial in [Greene, Curtis ; Zaslavsky, Thomas: On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs. In: Transactions of the American Mathematical Society 280 (1983), jan, Nr. 1, S. 97-97.]. We shall develop an invo…
Divisors on graphs, orientations, syzygies, and system reliability
我们研究了系统可靠性理论中出现的各种理想。我们使用的思想，从理论的因子，方向和拟阵图来描述这些理想的最小多面体细胞自由分解。在每种情况下，我们给出了一个明确的组合描述的最小生成集的每个较高的合合模的无环方向的图形，$q$ -简化除子和有界区域的图形超平面安排。这些理想的分解是密切相关的，它们的贝蒂数与基域的特征无关。我们应用这些结果来计算其相关的协调系统的可靠性。
Lower bounds on the maximum number of non-crossing acyclic graphs
本文是对点集上几何图计数问题的一个贡献。更具体地说，我们研究非交叉生成树和森林的最大数量。我们证明了N个点的所谓双链点配置具有Omega（12.52^N）非交叉生成树和Omega（13.61^N）非交叉森林。这改进了Dumitrescu，Schulz，Sheffer和Tóth给出的一般位置上所有N点集之间的非交叉生成树和非交叉森林的最大数目的下界.我们的分析依赖于分析组合学的工具，这使我们能够计算某些家庭的森林在凸位置的点，并估计其平均数量的组件。本文还得到了双链非交叉生成树个数的一个新的O（22.12 ^N）的上界.
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