Kassie Archer, Ira M. Gessel, Christina Graves, Xuming Liang
A descent of a labeled digraph is a directed edge (s, t) with s > t. We count strong tournaments, strong digraphs, and acyclic digraphs by descents and edges. To count strong tournaments we use Eulerian generating functions and to count strong and acyclic digraphs we use a new type of generating function that we call a graphic Eulerian generating function.
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A descent of a labeled acyclic digraph is a directed edge $x\to y$ with $x>y$. In this paper, we find a recurrence for the number of labeled acyclic digraphs with a given number of descents.
Acyclic Subgraphs of Planar Digraphs
有向图中的无圈集是指导出无圈子图的顶点集。Harutyunyan在2011年证明了$n$顶点上不含有向2-圈的平面有向图都有一个大小至少为$3n/5$的无圈集.我们证明了这个猜想的有向图，其中每个有向圈的长度至少为8。更一般地说，如果$g$是最短的有向循环的长度，我们表明，存在一个非循环集的大小至少为$(1 - 3/g)n$。
Let $D=(V,A)$ be an acyclic digraph. For $x\in V$ define $e_{_{D}}(x)$ to be the difference of the indegree and the outdegree of $x$. An acyclic ordering of the vertices of $D$ is a one-to-one map $g: V \rightarrow [1,|V|] $ that has the property that for all $x,y\in V$ if $(x,y)\in A$, then $g(x) < g(y)$. We prove that for every acyclic ordering $g$ of $D$ the following inequality holds: \[\sum_{x\in V} e_{_{D}}(x)\cdot g(x) ~\geq~ \frac{1}{2} \sum_{x\in V}[e_{_{D}}(x)]^2~.\] The class of acyclic digraphs for which equality holds is determined as the class of comparbility digraphs of posets o…
A matrix for counting paths in acyclic colored digraphs
本文提出一种与k-着色无环有向图关联的矩阵，并证明其行列式可枚举图中所有路径，从而建立矩阵行列式与路径计数之间的对应关系。
Acyclic Digraphs and Eigenvalues of (0,1)-Matrices
证明了具有n个标号顶点的无圈有向图的个数等于特征值为正实数的n × n（0，1）-矩阵的个数。
Acyclic and complete coloring of digraphs with the minimum and maximum possible numbers of colors
本文研究有向图无环与完全着色中颜色数的极值问题，构造了任意给定二色数 r 和双色数 t（r≤t）的非对称有向图，并给出非对称情形下的二次上界。
Vertex coloring acyclic digraphs and their corresponding hypergraphs
我们考虑了无圈有向图$\Gdag$的顶点着色，使得在$\Gdag$中有共同祖先的两个顶点得到不同的颜色.当为各种遗传数据限定空间以进行有效分析时，这种着色以自然的方式出现。我们讨论了相应的{ \em下色数}，并导出了一个上界作为$D(\Gdag)$、给定顶点的最大后代数和相应超图的退化度的函数。最后，我们确定了一个由$\Gdag$和$D(\Gdag)$的顶点数表示的下色数的渐近紧上界。
Large flames in rooted acyclic digraphs without backward-infinite paths
研究了无反向无限路径的无限有向图中火的性质，证明了Szeszlér定理的扩展形式及Lovász定理的无限推广
Thomassen's theorem on the two-linkage problem in acyclic digraphs: a shorter proof
该研究针对无环有向图的两链接问题，给出了Thomassen 1985年提出的经典定理的更简短证明，该定理刻画了无环有向图中不存在两条指定不相交有向路径的充要条件。
I present an algorithm that, given a number $n \geq 1$, computes a compact representation of the set of all noncrossing acyclic digraphs with $n$ nodes. This compact representation can be used as the basis for a wide range of dynamic programming algorithms on these graphs. As an illustration, along with this note I am releasing the implementation of an algorithm for counting the number of noncrossing acyclic digraphs of a given size. The same tabulation can be modified to count other classes of combinatorial structures, including weakly connected noncrossing acyclic digraphs, general noncrossi…
We prove that for every digraph $D$ and every choice of positive integers $k$, $\ell$ there exists a digraph $D^*$ with girth at least $\ell$ together with a surjective acyclic homomorphism $ψ\colon D^*\to D$ such that: (i) for every digraph $C$ of order at most $k$, there exists an acyclic homomorphism $D^*\to C$ if and only if there exists an acyclic homomorphism $D\to C$; and (ii) for every $D$-pointed digraph $C$ of order at most $k$ and every acyclic homomorphism $φ\colon D^*\to C$ there exists a unique acyclic homomorphism $f\colon D\to C$ such that $φ=f\circψ$. This implies the main res…
We consider the extension to directed graphs of the concept of achromatic number in terms of acyclic vertex colorings. The achromatic number have been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromatic number is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982. A coloring of a digraph is an acyclic coloring if each subdigraph induced by each chromatic class is acyclic, and a coloring is complete if for any pair of chromatic classes $x,y$, there is an arc from $x$ to $y$ and an arc from $y$ to $x$. The dichroma…
Hefner [K. A. S. Hefner, K. F. Jones, S. -R. Kim, R. J. Lundgren and F. S. Roberts: $(i,j)$ competition graphs, Discrete Applied Mathematics, 32, (1991) 241-262] characterized acyclic digraphs each vertex of which has inderee and outdegree at most two and whose competition graphs are interval. They called acyclic digraphs each vertex of which has inderee and outdegree at most two $(2,2)$ digraphs. In this paper, we study the phylogeny graphs of $(2,2)$ digraphs. Especially, we give a sufficient condition and necessary conditions for $(2,2)$ digraphs having chordal phylogeny graphs. Phylogeny g…
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 consider acyclic r-colorings in graphs and digraphs: they color the vertices in r colors, each of which induces an acyclic graph or digraph. (This includes the dichromatic number of a digraph, and the arboricity of a graph.) For any girth and sufficiently high degree, we prove the NP-completeness of acyclic r-colorings; our method also implies the known analogue for classical colorings. The proofs use high girth graphs with high arboricity and dichromatic numbers. High girth graphs and digraphs with high chromatic and dichromatic numbers have been well studied; we re-derive the results from…
The dichromatic number of a digraph $G$ is the smallest integer $χ_a(G)$ such that the vertex set of $G$ can be partitioned into $χ_a(G)$ sets, each of which induces an acyclic subdigraph. This is a generalization of the classic chromatic number of graphs. Here, we investigate the dichromatic number of the cartesian, direct, strong and lexicographic products, giving generalizations of some classic results on the chromatic number of products. More specifically, we prove that the following inequalities, known to hold for the chromatic number of graphs, still hold for the dichromatic number of di…
More on counting acyclic digraphs
本文通过对简单无圈有向图的递归公式的稍加修改，导出了几类标号无圈有向图的计数公式。这些考虑的动机，并基于最近的组合结果几何拓扑获得的S.崔，谁建立了确切的对应关系之间的无圈有向图和所谓的小覆盖超立方体和相关的多面体。特别是，我们证明了$r$ -单纯形的$n$副本的笛卡儿积上的小覆盖的等价类的数目等于无圈$(2^r-1)$ -重有向图的阶$n$的数目。由于主公式是用一个简单的方程表示的，所以渐近性很容易得到。
Real Bott manifolds and acyclic digraphs
Masuda（2008）通过对上三角形矩阵的三种运算给出了实Bott流形的特征。我们提供了一个组合特征的真正的Bott流形到同构的操作方面的有向无圈图。我们的观察导致几个新的不变量的真正的博特流形。
Burr and Erdős in 1975 conjectured, and Chvátal, Rödl, Szemerédi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed analogue of the Burr--Erdős conjecture, answering a question of Bucić, Letzter, and Sudakov. If $H$ is an acyclic digraph, the oriented Ramsey number of $H$, denoted $\overrightarrow{r_{1}}(H)$, is the least $N$ such that every tournament on $N$ vertices contains a copy of $H$. We show that for any $Δ\geq 2$ and any sufficiently large $n$, there exists an acyclic digraph…
An acyclic homomorphism of a digraph $C$ to a digraph $D$ is a function $ρ\colon V(C)\to V(D)$ such that for every arc $uv$ of $C$, either $ρ(u)=ρ(v)$, or $ρ(u)ρ(v)$ is an arc of $D$ and for every vertex $v\in V(D)$, the subdigraph of $C$ induced by $ρ^{-1}(v)$ is acyclic. A digraph $D$ is a core if the only acyclic homomorphisms of $D$ to itself are automorphisms. In this paper, we prove that for certain choices of $p(n)$, random digraphs $D\in D(n,p(n))$ are asymptotically almost surely cores. For digraphs, this mirrors a result from [A. Bonato and P. Prałat, The good, the bad, and the great…
Given a directed graph, an acyclic set is a set of vertices inducing a subgraph with no directed cycle. In this note we show that there exist oriented planar graphs of order $n$ for which the size of the maximum acyclic set is at most $\lceil \frac{n+1}{2} \rceil$, for any $n$. This disproves a conjecture of Harutyunyan and shows that a question of Albertson is best possible.
Coloring of a Digraph
\qquad有向图$D=(V,E)$的\emph{coloring}是其顶点的着色，遵循以下规则：令$uv$是$D$中的一条弧。如果尾部$u$先着色，则头部$v$的颜色应不同于$u$。$D$的\emph{dichromatic number} $χ_d(D)$是$D$着色中所需的最小颜色数。除了得到类似于图的色数的$χ_d(D)$的许多结果和界外，我们还证明了当$D$是无圈的时，$χ_d(D)=1$是有界的.引入了图/有向图序列着色的新概念。利用点标有向图的$L$ -矩阵给出了无圈有向图的一个刻划.
Cut covers of acyclic digraphs
该研究证明无环有向图割覆盖的入度条件是紧的，给出有向路径幂图割覆盖的最大幂次范围，并证明相关同态有向图的割覆盖判定问题是NP完全的。
Acyclic subgraphs of digraphs with high chromatic number
研究了具有高色数的有向图的无环子图色数，证明了在特定条件下该色数下界为$n^{5/9-o(1)}$，并改进了现有界。
Competition-common enemy graphs of degree-bounded digraphs
本文完整刻画了⟨2,2⟩有向图的竞争-公共敌图（CCE图），证明分量数≤7的无环⟨2,2⟩有向图的CCE图为区间图且界紧，拓展了度有界有向图相关图类的研究。
Directed acyclic graphs (DAGs) can be characterised as directed graphs whose strongly connected components are isolated vertices. Using this restriction on the strong components, we discover that when $m = cn$, where $m$ is the number of directed edges, $n$ is the number of vertices, and $c < 1$, the asymptotic probability that a random digraph is acyclic is an explicit function $p(c)$, such that $p(0) = 1$ and $p(1) = 0$. When $m = n(1 + μn^{-1/3})$, the asymptotic behaviour changes, and the probability that a digraph is acyclic becomes $n^{-1/3} C(μ)$, where $C(μ)$ is an explicit function of…
Acyclic Subgraphs in $k$-Majority Tournaments
一个$k$ -多数有向图是一个有向图，它是通过在同一个基集上组合$k$个体排序来形成一个共识，其中边指向由严格多数排序指示的方向。$k$ -多数有向图用于模拟投票场景，其中顶点对应于由$k$投票者排序的选项。当$k$是奇数时，所得到的有向图总是一个竞赛图，称为$k$ -多数竞赛图。设$f_k(n)$是所有具有$n$顶点的$k$ -多数竞赛图中导出的传递子竞赛图的最大阶的最小值.最近，Milans、Schreiber和West证明了$\sqrt n \le f_3(n) \le 2 \sqrt n +1 $。本文改进了$f_3(n)$的上界，证明了$f_3(n) < \sqrt {2n} +\frac 12 $ .
We investigate the complexity of generalizations of colourings (acyclic colourings, $(k,\ell)$-colourings, homomorphisms, and matrix partitions), for the class of transitive digraphs. Even though transitive digraphs are nicely structured, many problems are intractable, and their complexity turns out to be difficult to classify. We present some motivational results and several open problems.
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。
Orthogonality between acyclic subdigraphs and paths in digraphs
本文研究了有向图中无环子图与路径的正交性，证明了在替换稳定集为诱导无环子图后，可以放松两个开放猜想。
Generalized tableaux over arbitrary digraphs and their associated differential equations
本文在任意多重有向图上推广无环序的排列与计数器，证明嵌套有向路径下计数器生成函数为多项式与指数函数之积，并借助两行排列有向图构造出具有多项式解的低阶线性微分方程，特别得到与 Catalan 数相关的 Catalan 微分方程和正交 Catalan 多项式。
Improved lower bounds for the maximum order of an induced acyclic subgraph
本文改进了有向图最大诱导无环顶点集大小的下界，通过结合局部度数数据和反馈顶点集方差分析，提出更精确的下界方法。
Extensions of results on phylogeny graphs of degree bounded digraphs
该研究延续贝叶斯网络证据传播相关的系统发育图研究方向，扩展了已有成果，给出弦$(i,2)$系统发育图的必要条件，并通过列出禁用诱导子图给出$(i,j)$系统发育图的必要条件。
Let $D$ be a digraph and let $λ(D)$ denote the number of vertices in a longest path of $D$. For a pair of vertex-disjoint induced subdigraphs $A$ and $B$ of $D$, we say that $(A,B)$ is a partition of $D$ if $V(A)\cup V(B)=V(D).$ The Path Partition Conjecture (PPC) states that for every digraph, $D$, and every integer $q$ with $1\leq q\leqλ(D)-1$, there exists a partition $(A,B)$ of $D$ such that $λ(A)\leq q$ and $λ(B)\leqλ(D)-q.$ Let $T$ be a digraph with vertex set $\{u_1,\dots, u_t\}$ and for every $i\in [t]$, let $H_i$ be a digraph with vertex set $\{u_{i,j_i}\colon\, j_i\in [n_i]\}$. The {…
Blow-ups and extensions of trees in tournaments
本研究将有向树的线性不可避免性推广，证明了添加至多k个通用顶点的有向树及有向树的k-膨胀图均为线性不可避免，并给出了对应的竞赛图阶数界。
Ramsey numbers of digraphs with local edge structure
该研究针对有向图的Ramsey数线性性问题，提出分级带宽刻画局部边结构，证明有界度无环且分级带宽有界的有向图定向Ramsey数线性，统一推广了多项已有结果。
Minimal Path and Acyclic Models in the Path Complex
本文研究有向图路径复形的极小路径结构，分析极小3-路径实例，证明极小路径支撑子有向图具有无环路径同调，并探讨该无环模型的应用。
The chromatic number of a digraph $D$ is the minimum number of acyclic subgraphs covering the vertex set of $D$. A tournament $H$ is a hero if every $H$-free tournament $T$ has chromatic number bounded by a function of $H$. Inspired by the celebrated Erdős--Hajnal conjecture, Berger et al. fully characterized the class of heroes in 2013. We extend this framework to dense digraphs: A digraph $H$ is a superhero if every $H$-free digraph $D$ has chromatic number bounded by a function of $H$ and $α(D)$, the independence number of the underlying graph of $D$. We prove here that a digraph is a super…
Classification of real Bott manifolds and acyclic digraphs
本文通过矩阵运算和图运算刻画实Bott流形的仿射微分同胚分类，证明其乘积分解唯一性及环面秩猜想成立。
In 1982 Víctor Neumann-Lara introduced the dichromatic number of a digraph $D$ as the smallest integer $k$ such that the vertices $V$ of $D$ can be colored with $k$ colors and each color class induces an acyclic digraph. Later a flow theory for the dichromatic number transferring Tutte's theory of nowhere-zero flows (NZ-flows) from classic graph colorings has been developed by Hochstättler. The purpose of this paper is to pursue this analogy by introducing a new definition of algebraic Neumann-Lara-flows (NL-flows) and a closed formula for their polynomial. Furthermore we generalize the Equiva…
The Path Partition Conjecture and 3-(quasi-)Transitive Digraphs
本文针对3-传递和3-拟传递有向图，引入半无环有向图概念刻画其结构，并将路划分猜想的成立范围推广至扩展半无环有向图类。
An equality for balanced digraphs
本文证明平衡有向图中，不含环且每个顶点到指定顶点s有路径的k边子集（称为s-收敛）的数量与s无关，推广了关于生成树形图、无环定向和最大无环子图的结果。
Topological Additive Numbering of Directed Acyclic Graphs
我们建议研究一个自然产生的问题，从两个拓扑编号的有向无环图，和添加剂着色（也称为幸运标签）。设$D$是一个有向图，$f$是它的顶点的正整数标号，用$S(v)$表示每个顶点$v$的所有邻点上的标号之和.如果有向图的每个弧$(u,v)$的$S(u) < S(v)$为，则标记$f$称为\emph{topological additive numbering}。该问题要求找到$D$具有拓扑加法编号的最小数量$k$，其中标签属于$\{ 1, \ldots, k \}$，由$η_t(D)$表示。我们刻画了一个有向图的拓扑加法数，给出了$η_t(D)$的一个下界，并给出了一个整数规划公式，刻画了当它的系数矩阵是完全幺模的。我们还提出了一些家庭的$η_t(D)$可以计算在多项式时间。最后，我们证明了这个问题是\np -硬，即使它的输入是限制在平面二部有向图。
The \textit{acyclic disconnection} $\overrightarrow{ω}(D)$ (resp. the \textit{directed triangle free disconnection } $\overrightarrow{ω}_{3}(D)$) of a digraph $D$ is defined as the maximum possible number of connected components of the underlying graph of $D\setminus A(D^{\ast })$ where $D^{\ast }$ is an acyclic (resp. a directed triangle free) subdigraph of $D$. In this paper, we generalize some previous results and solve some problems posed by V. Neumann-Lara (The acyclic disconnection of a digraph, Discrete Math. 197/198 (1999), 617-632). Let $\overrightarrow{C}_{2n+1}(J)$ be a circulant to…
Acyclic dichromatic number of oriented graphs
本文研究了有向图的无环二色数，定义并分析了其与传统二色数的区别，证明了在某些图中无环二色数可能远大于传统二色数，并展示了相关复杂性结果。
Let $T$ be a digraph with vertices $u_1, \dots, u_t$ ($t\ge 2$) and let $H_1, \dots, H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1, \dots, H_t]$ is a digraph with vertex set $\{u_{i,j_i}\colon\, 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\colon\, u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ The composition $Q=T[H_1, \dots, H_t]$ is a semicomplete composition if $T$ is semicomplete, i.e. there is at least one arc between every pair of vertices. Digraph compositions generalize 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。
Eulerian Polynomials for Digraphs
本文研究有向图标号下降的生成函数欧拉多项式，受 Kalai 工作启发，针对底图为二部图的情形，给出了该多项式在 -1 处绝对值的广义交替排列组合解释。
In this paper, we compute competition indices and periods of multipartite tournaments. We first show that the competition period of an acyclic digraph $D$ is one and $ζ(D) +1$ is a sharp upper bound of the competition index of $D$ where $ζ(D)$ is the sink elimination index of $D$. Then we prove that, especially, for an acyclic $k$-partite tournament $D$, the competition index of $D$ is $ζ(D)$ or $ζ(D) +1$ for an integer $k \ge 3$. By developing useful tools to create infinitely many directed walks in a certain regular pattern from given directed walks, we show that the competition period of a …
The Redei-Berge Hopf algebra of digraphs
本文证明Richard Stanley提出的Redei-Berge对称函数源自有向图上的组合Hopf代数结构，导出的多项式满足删除-收缩性质，且Berge关于哈密顿路径的经典结果是其互反公式的推论。
Dichromatic Number and Cycle Inversions
本文用Farkas引理简洁证明了有向图无环色数的循环排序刻画，并表明通过逐次反转一族有向循环可将其降至至多2。
Stanley defined the chromatic symmetric function of a graph, and Shareshian and Wachs introduced a refinement, namely the chromatic quasisymmetric function of a labeled graph. In this paper, we define the chromatic quasisymmetric function of a directed graph, which agrees with the Shareshian-Wachs definition in the acyclic case. We give an F-basis expansion for all digraphs in terms of a permutation statistic, which we call G-descents. We use this expansion to derive a p-positivity formula for all digraphs with symmetric chromatic quasisymmetric functions. We show that the chromatic quasisymme…
Let $G$ be a graph of order $n$. It is well-known that $α(G)\geq \sum_{i=1}^n \frac{1}{1+d_i}$, where $α(G)$ is the independence number of $G$ and $d_1,\ldots,d_n$ is the degree sequence of $G$. We extend this result to digraphs by showing that if $D$ is a digraph with $n$ vertices, then $ α(D)\geq \sum_{i=1}^n \left( \frac{1}{1+d_i^+} + \frac{1}{1+d_i^-} - \frac{1}{1+d_i}\right)$, where $α(D)$ is the maximum size of an acyclic vertex set of $D$. Golowich proved that for any digraph $D$, $χ(D)\leq \lceil \frac{4k}{5} \rceil+2$, where $k=max(Δ^+(D),Δ^-(D))$. We give a short and simple proof for…
The number of small covers over cubes
本文给出了$n$ -立方体上的小覆盖集与具有$n$标号结点的无圈有向图集之间的一个双射。利用这一点，我们给出了$n$ -立方体（一般是单形的乘积）上直到Davis-Januszkiewicz等价类和$\mathbf{Z}^n$ -等变同形类的小覆盖数的一个公式.此外，我们还证明了$n$无标号节点的无圈有向图的个数是$n$ -立方体上小覆盖个数的上界，直到同构.
Towards Transitive-free Digraphs
本文研究 $\ell$-RTVD 问题，在多类有向图上给出多项式可解、NP 难及参数化核与不可处理性结果。
Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurović et al. (IEEE TCBB, to appear) introduced the minimum conflict-free row split (MCRS) problem: split each row of a given binary matrix into a bitwise OR of a set of rows so that the resulting matrix corresponds to a perfect phylogeny and has the minimum possible number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, in which the task is to minimize the number of distinct rows of the resulting matrix. Hujdur…
Acyclic Dichromatic Number of Tournaments: these are the Champions
研究竞赛图的无圈双色数，刻画了有足够大无圈双色数的竞赛图中必出现的子竞赛图，证实相关猜想，证明其满足局部到全局性质。
Bounding the chromatic number of dense digraphs by arc neighborhoods
本文证明竞赛图中所有弧邻域色数有界则整图色数有界，并推广到独立数有界的稠密定向图，由此建立两个高色数结构猜想的等价性。
Heroes in oriented complete multipartite graphs
该研究针对有向图的英雄图刻画问题，证明拟传递有向图的英雄图与竞赛图一致，证否了有向多部图相关猜想，并给出有向完全多部图英雄图的近乎完整刻画。
In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equ…
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