Gerard Cohen, Emanuela Fachini, Janos Korner
We extend several results of the third author and C. Malvenuto on graph-different permutations to the case of directed graphs and introduce new open problems. Permutation capacity is a natural extension of Sperner capacity from finite directed graphs to infinite digraphs. Our subject is combinatorial in nature, but can be equally regarded as zero-error information theory.
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A permutation in a digraph $G=(V, E)$ is a bijection $f:V \rightarrow V$ such that for all $v \in V$ we either have that $f$ fixes $v$ or $(v, f(v)) \in E$. A derangement in $G$ is a permutation that does not fix any vertex. In [1] it is proved that in any digraph, the ratio of derangements to permutations is at most $1/2$. Answering a question posed in [1], we show that the set of possible ratios of derangements to permutations in digraphs is dense in the interval $[0, 1/2]$.
A structure $\cal S$ is quasi-projective if for every structure $\cal T$, for every homomorphism $f : {\cal S} \rightarrow {\cal T}$ and every epimorphism $j: {\cal S}\rightarrow {\cal T}$ there is an endomorphism $ϕ$ of $\cal S$ such that $ϕ\circ j=f$. In this paper, we characterise the quasi-projective posets and lattices of arbitrary cardinalities, finite permutations, graphs and digraphs of arbitrary cardinalities with loops and without loops, finite hypergraphs, and finite point-line geometries.
If $G$ is a group acting on a set $Ω$ and $α, β\in Ω$, the digraph whose vertex set is $Ω$ and whose arc set is the orbit $(α, β)^G$ is called an {\em orbital digraph} of $G$. Each orbit of the stabiliser $G_α$ acting on $Ω$ is called a {\it suborbit} of $G$. A digraph is {\em locally finite} if each vertex is adjacent to at most finitely many other vertices. A locally finite digraph $Γ$ has more than one end if there exists a finite set of vertices $X$ such that the induced digraph $Γ\setminus X$ contains at least two infinite connected components; if there exists such a set containing precis…
Hamiltonian paths in the permutation digraphs $P(n,n-2)$
该研究针对Isaak提出的问题，证明了$n\geq4$时无有向哈密顿环的置换有向图$P(n,n-2)$存在有向哈密顿路径，给出了肯定答案。
Promotion digraphs
研究为任意形状的标准和递增表格引入提升有向图，证明任意形状标准表格可由其唯一确定，猜测矩形递增表格也如此并给出证据，还用三行矩形递增表格的提升有向图猜测其与火烈鸟网的联系。
We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees. We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye in 1998. Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes. For complete binary trees we show that as…
The characteristic polynomial of sums of random permutations and regular digraphs
本文研究随机置换矩阵之和 $A_n$ 的归一化特征多项式，证明其收敛于单位圆盘上的随机解析函数，并据此给出随机正则有向图谱隙的初等证明。
Counting fixed-point-free Cayley permutations
通过双种类方法得到Cayley排列的函数有向图的微分方程，推导出无不动点Cayley排列的显式公式，并证明其比例趋于1/e，与排列和自映射一致。
Continued fractions using a Laguerre digraph interpretation of the Foata--Zeilberger bijection and its variants
本文用 Laguerre 有向图解释 Foata--Zeilberger 双射以计算排列中的环，从而得到了枚举排列和 D-排列的多变量多项式的 Jacobi 型、Thron 型和 Stieltjes 型连分数，并证明了 Sokal、Zeng 等人提出的连分数猜想。
Mahonian-Stirling statistics for partial permutations
本文为部分排列构造了与逆序数等分布的 MacMahon 主指标类比，建立 Haglund-Remmel-Wilson 型恒等式，并用 Laguerre 有向图推广了 Jacobi-Rogers 多项式的组合解释。
We characterise the primitive 2-closed groups $G$ of rank at most four that are not the automorphism group of a graph or digraph and show that if the degree is at least 2402 then there are just two infinite families or $G\leqslant \mathrm{A}Γ\mathrm{L}_1(p^d)$, the 1-dimensional affine semilinear group. These are the first known examples of non-regular 2-closed groups that are not the automorphism group of a graph or digraph.
Let $Γ$ be a finite $G$-vertex-transitive digraph. The in-local action of $(Γ,G)$ is the permutation group $L_-$ induced by the vertex-stabiliser on the set of in-neighbours of $v$. The out-local action $L_+$ is defined analogously. Note that $L_-$ and $L_+$ may not be isomorphic. We thus consider the problem of determining which pairs $(L_-,L_+)$ are possible. We prove some general results, but pay special attention to the case when $L_-$ and $L_+$ are both quasiprimitive. (Recall that a permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive.) Along the w…
Quasi-strongly regular digraphs constructed from transitive groups of degree $n\leq 110$
本文提出一种从传递置换群构造有向正则图的推广方法，证明多组参数的有向强正则图存在，并分类了低次数传递群和本原群产生的拟强正则有向图。
The general position number of digraphs
研究有向图中最大无三顶点共最短路径的顶点集，证明问题在定向图上为NP难，并探讨了循环图、卡茨图和排列图等重要类别的通用位置数。
A transitive permutation group is called elusive if it contains no semiregular element. We show that no group of automorphisms of a connected graph of valency at most four is elusive and determine all the elusive groups of automorphisms of connected digraphs of out-valency at most three.
This version is similar to math.CO/0210113. We've changed Conjectures 1.1 and 1.2 so that they cover arbitrary graphs(digraphs). Let G be an arbitrary graph(digraph). Then - in polynomial time - either an algorithm obtains a hamilton circuit(cycle)or else the algorithm points to at least one vertex that cannot belong to any hamilton circuit(cycle) of G. We give criteria for determining which vertices should be examined.
Algebraic Constructions for the Digraph Routing Problems
该研究利用拟群、置换群等代数工具研究正则有向图路由的最优性质，给出重要图的紧凑代数定义，并发现广群可由不相交置换唯一表示。
New 2-closed groups that are not automorphism groups of digraphs
本研究拓展已有构造方法，首次发现秩大于4的非正则2-闭置换群不能作为任何有向图的自同构群，并证明该构造仅适用于四个特定素数，丰富了置换群与图自同构群的相关研究。
We define strongly chordal digraphs, which generalize strongly chordal graphs and chordal bipartite graphs, and are included in the class of chordal digraphs. They correspond to square 0,1 matrices that admit a simultaneous row and column permutation avoiding the Γ matrix. In general, it is not clear if these digraphs can be recognized in polynomial time, and we focus on symmetric digraphs (i.e., graphs with possible loops), tournaments with possible loops, and balanced digraphs. In each of these cases we give a polynomial-time recognition algorithm and a forbidden induced subgraph characteriz…
Non-Hamiltonian 2-regular Digraphs -- Residues
本文研究非哈密顿2-正则有向图的性质，通过引入双共轭、排除集和余数的概念，证明了非哈密顿性的必要且充分条件。
In the paper Graphical complexity of products of permutation groups, M. Grech, A. Jez, A. Kisielewicz have proved that the direct product of automorphism groups of edge-colored graphs is itself the automorphism groups of an edge-colored graph. In this paper, we study the direct product of two permutation groups such that at least one of them fails to be the automorphism group of an edge-colored graph. We find necessary and sufficient conditions for the direct product to be the automorphism group of an edge-colored graph. The same problem is solved for the edge-colored digraphs.
Steinerberger Curvature On Digraphs -- Discrete Bonnet-Myers and Lichnerowicz Theorems
研究通过平衡测度编码图全局距离几何的施泰纳伯格曲率，推导二面体群和广义四元数群无向凯莱图的曲率公式，扩展到强连通简单有向图，建立其结构性质及相关定理。
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…
Some families of digraphs determined by the complementarity spectrum
本文研究互补谱区分非同构有向图的能力，聚焦于恰好具有三个互补特征值的七个族，发现部分族中互补谱可唯一确定图，并优于传统特征值，为有向图结构提供新信息。
The zero-error capacity of channels with a countably infinite input alphabet formally generalises Shannon's classical problem about the capacity of discrete memoryless channels. We solve the problem for three particular channels. Our results are purely combinatorial and in line with previous work of the third author about permutation capacity.
An intersection digraph is a digraph where every vertex $v$ is represented by an ordered pair $(S_v, T_v)$ of sets such that there is an edge from $v$ to $w$ if and only if $S_v$ and $T_w$ intersect. An intersection digraph is reflexive if $S_v\cap T_v\neq \emptyset$ for every vertex $v$. Compared to well-known undirected intersection graphs like interval graphs and permutation graphs, not many algorithmic applications on intersection digraphs have been developed. Motivated by the successful story on algorithmic applications of intersection graphs using a graph width parameter called mim-width…
Black-White Cell Capacity in $k$-ary Words and Permutations
本文提出条形图新统计量黑白格容量，研究 $k$ 元词与排列中黑白格数量分布，给出生成函数、闭式公式及渐近，并特别关注黑白平衡词。
We investigate a communication setup where a source output is sent through a free noisy channel first and an additional codeword is sent through a noiseless but expensive channel later. With the help of the second message the decoder should be able to decide with zero-error whether its decoding of the first message was error-free. This scenario leads to the definition of a digraph parameter that generalizes Witsenhausen's zero-error rate for directed graphs. We investigate this new parameter for some specific directed graphs and explore its relations to other digraph parameters like Sperner ca…
We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results in case of complementary graphs. We also explore the close relationship between our problem and the concept of Shannon capacity "within a given type".
In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to th…
The modular decomposition of a graph is a canonical representation of its modules. Algorithms for computing the modular decomposition of directed and undirected graphs differ significantly, with the undirected case being simpler, and algorithms for directed graphs often work by reducing the problem to decomposing undirected graphs. In this paper we show that transitive acyclic digraphs have the same strong modules as their undirected versions. This simplifies reduction for transitive digraphs, requiring only the computation of strongly connected components. Furthermore, we are interested in pe…
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。
A Boolean network (BN) with $n$ components is a discrete dynamical system described by the successive iterations of a function $f:\{0,1\}^n\to\{0,1\}^n$. In most applications, the main parameter is the interaction graph of $f$: the digraph with vertex set $\{1,\dots,n\}$ that contains an arc from $j$ to $i$ if $f_i$ depends on input $j$. What can be said on the set $\mathcal{G}(f)$ of the interaction graphs of the BNs $h$ isomorphic to $f$, that is, such that $h\circ π=π\circ f$ for some permutation $π$ of $\{0,1\}^n$? It seems that this simple question has never been studied. Here, we report …
A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian; this baseline result is used as the basis of existence proofs for universal cycles (also known as ucycles or generalized deBruijn cycles or U-cycles) of several combinatorial objects. The existence of ucycles is often dependent on the specific representation that we use for the combinatorial objects. For example, should we represent the subset $\{2,5\}$ of $\{1,2,3,4,5\}$ as "25" in a linear string? Is the representation "52" acceptable? Or it it tactically advantageous (and acceptable) to go with $\{0,…
Let D be an arbitrary subset of the natural numbers. For every n, let M(n;D) be the maximum of the cardinality of a set of Hamiltonian paths in the complete graph K_n such that the union of any two paths from the family contains a not necessarily induced cycle of some length from D. We determine or bound the asymptotics of M(n;D) in various special cases. This problem is closely related to that of the permutation capacity of graphs and constitutes a further extension of the problem area around Shannon capacity. We also discuss how to generalize our cycle-difference problems and present an exam…
Linear extensions and directed clique counts via modular partitions
本文提出通过模分区结构递归计算偏序集线性扩展的新方法，并探讨其在有向图和神经回路中的应用。
Counting Kernels in Directed Graphs with Arbitrary Orientations
该研究针对有向图核计数与判定的NP难问题，提出了不依赖边定向的模糊圆形区间图核多项式计数与存在时构造算法，并证明补可约图核问题一般NP难但阈值图子类可线性时间求解。
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