Stephen C. Locke, Dave Witte Morris
We construct infinitely many connected, circulant digraphs of outdegree three that have no hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is adjacent to two diametrically opposite vertices, or every vertex is adjacent to the vertex diametrically opposite to itself.
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Mader conjectured that for all k there is an integer d(k) such that every digraph of minimum outdegree at least d(k) contains a subdivision of a transitive tournament of order k. In this note we observe that if the minimum outdegree of a digraph is sufficiently large compared to its order then one can even guarantee a subdivision of a large complete digraph. More precisely, let G be a digraph of order n whose minimum outdegree is at least d. Then G contains a subdivision of a complete digraph of order at least d^2/(8n^{3/2}).
We prove the existence of a function $h(k)$ such that every simple digraph with minimum outdegree greater than $h(k)$ contains an immersion of the transitive tournament on $k$ vertices. This solves a conjecture of Devos, McDonald, Mohar and Scheide.
Subgraphs with a positive minimum semidegree in digraphs with large outdegree
该研究证明了在具有大出度的有向图中存在一个子图，其最小半度数满足特定下界，并展示了该界在特定条件下是渐近最优的。
In 1985, Mader conjectured that for every acyclic digraph $F$ there exists $K=K(F)$ such that every digraph $D$ with minimum out-degree at least $K$ contains a subdivision of $F$. This conjecture remains widely open, even for digraphs $F$ on five vertices. Recently, Aboulker, Cohen, Havet, Lochet, Moura and Thomassé studied special cases of Mader's problem and made the following conjecture: for every $\ell \geq 2$ there exists $K = K(\ell)$ such that every digraph $D$ with minimum out-degree at least $K$ contains a subdivision of every orientation of a cycle of length $\ell$. We prove this con…
Scott asked the question of determining $c_d$ such that if $D$ is a digraph with $m$ arcs and minimum outdegree $d\ge 2$ then $V(D)$ has a partition $V_1, V_2$ such that $\min\left\{e(V_1,V_2),e(V_2, V_1)\right\}\geq c_dm$, where $e(V_1,V_2)$ (respectively, $e(V_2,V_1)$) is the number of arcs from $V_1$ to $V_2$ (respectively, from $V_2$ to $V_1$). Lee, Loh, and Sudakov showed that $c_2=1/6+o(1)$ and $c_3=1/5+o(1)$, and conjectured that $c_d= \frac{d-1}{2(2d-1)}+o(1)$ for $d\ge 4$. In this paper, we show $c_4=3/14+o(1)$ and prove some partial results for $d\ge 5$.
We classify certain non-symmetric commutative association schemes. As an application, we determine all the primitive weakly distance-regular circulant digraphs.
Bermond and Thomassen conjectured that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex disjoint cycles. So far the conjecture was verified for $k\le 3$. Here we generalise the question asking for all outdegree sequences which force $k$ vertex disjoint cycles and give the full answer for $k\le 2$.
Let $G=(V(G) ,E(G))$ be a digraph without loops and multiarcs, where $V(G)=\{v_1,v_2,\ldots,v_n\}$ and $E(G)$ are the vertex set and the arc set of $G$, respectively. Let $d_i^{+}$ be the outdegree of the vertex $v_i$. Let $A(G)$ be the adjacency matrix of $G$ and $D(G)=\textrm{diag}(d_1^{+},d_2^{+},\ldots,d_n^{+})$ be the diagonal matrix with outdegrees of the vertices of $G$. Then we call $Q(G)=D(G)+A(G)$ the signless Laplacian matrix of $G$. The spectral radius of $Q(G)$ is called the signless Laplacian spectral radius of $G$, denoted by $q(G)$. In this paper, some upper bounds for $q(G)$ a…
We consider the problem of immersing the complete digraph on t vertices in a simple digraph. For Eulerian digraphs, we show that such an immersion always exists whenever minimum degree is at least t(t-1), and for t at most 4 minimum degree at least t-1 suffices. On the other hand, we show that there exist non-Eulerian digraphs with all vertices of arbitrarily high in- and outdegree which do not contain an immersion of the complete digraph on 3 vertices. As a side result, we obtain a construction of digraphs with large outdegree in which all cycles have odd length, simplifying a former construc…
Donald Knuth, in a draft of a coming volume of The Art of Computer Programming, has recently conjectured that in Depth-First Search of a random digraph with geometric outdegree distribution, the numbers of back and forward arcs have the same distribution. We show that this conjecture is equivalent to an equality between two generating functions defined by different recursions. Unfortunately, we have not been able so use this to prove the conjecture, which still is open, but we hope that this note will inspire others to succeed with the conjecture.
Given a positive integer $m\ge 3$, let $ch(m)$ be the smallest positive constant with the following property: \emph{ Every simple directed graph on $n\ge 3$ vertices all whose outdegrees are at least $ch(m)\cdot n$ contains a directed cycle of length at most $m$.} Caccetta and Häggkvist conjectured that $ch(m)=1/m$, which if true, would be the best possible. In this paper, we prove the following result: \emph{ For every integer $m\ge 3$, let $α(m)$ be the unique real root in $(0,1)$ of the equation} \begin{equation*} (1-x)^{m-2}=\frac{3x}{2-x}. \end{equation*} Then $ch(m)\le α(m)$. This genera…
Let D be the circulant digraph with n vertices and connection set {2,3,c}. (Assume D is loopless and has outdegree 3.) Work of S.C.Locke and D.Witte implies that if n is a multiple of 6, c is either (n/2) + 2 or (n/2) + 3, and c is even, then D does not have a hamiltonian cycle. For all other cases, we construct a hamiltonian cycle in D.
A classification of locally-quasiprimitive circulant digraphs
研究局部拟本原循环有向图的分类，通过扩展局部本原循环图分类，确定其同构类型包括完全图、完全二部图等多种图，给出了连通局部拟本原循环有向图的具体分类结果。
A weighted digraph is a digraph such that every arc is assigned a nonnegative number, called the weight of the arc. The weighted outdegree of a vertex $v$ in a weighted digraph $D$ is the sum of the weights of the arcs with $v$ as their tail, and the weight of a directed cycle $C$ in $D$ is the sum of the weights of the arcs of $C$. In this note we prove that if every vertex of a weighted digraph $D$ with order $n$ has weighted outdegree at least 1, then there exists a directed cycle in $D$ with weight at least $1/\log_2 n$. This proves a conjecture of Bollobás and Scott up to a constant facto…
We show that almost all circulant graphs have automorphism groups as small as possible. Of the circulant graphs that do not have automorphism group as small as possible, we give some families of integers such that it is not true that almost all circulant graphs whose order lies in any one of these families, are normal. That almost all Cayley (di)graphs whose automorphism group is not as small as possible are normal was conjectured by the second author, so these results provide counterexamples to this conjecture. It is then shown that there is a large family of integers for which almost every c…
In this paper, we determine the achromatic and diachromatic numbers of some circulant graphs and digraphs each one with two lengths and give bounds for other circulant graphs and digraphs with two lengths. In particular, for the achromatic number we state that $α(C_{16q^2+20q+7}(1,2))=8q+5$, and for the diachromatic number we state that $dac(\overrightarrow{C}_{32q^2+24q+5}(1,2))=8q+3$. In general, we give the lower bounds $α(C_{4q^2+aq+1}(1,a))\geq 4q+1$ and $dac(\overrightarrow{C}_{8q^2+2(a+4)q+a+3}(1,a))\geq 4q+3$ when $a$ is a non quadratic residue of $\mathbb{Z}_{4q+1}$ for graphs and $\m…
Let $k$ be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles. It is famous as one of the one hundred unsolved problems selected in [Bondy, Murty, Graph Theory, Springer-Verlag London, 2008]. Lichiardopol, Por and Sereni proved in [SIAM J. Discrete Math. 23 (2) (2009) 979-992] that the above conjecture holds for $k=3$. Let $g$ be the girth, i.e., the length of the shortest cycle, of a given digraph. Bang-Jensen, Bessy and Thomassé conjectured in [J. Graph Theory 75 (3) (2014) 284-302] that …
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…
Ordering digraphs with maximum outdegrees by their $A_α$ spectral radius
本文研究强连通有向图的 $A_α$ 谱半径排序，通过给出两个上界，证明在最大出度满足特定条件时，具有更大最大出度的有向图具有更大的 $A_α$ 谱半径（或相差不超过 $\frac{1}{4}$）。
On Link-irregular Digraphs
研究了链接不规则有向图的性质，证明了其存在条件，并构造了相关例子。
A note on three-quarters circulant digraphs
本文引入一类与循环群相关的四分之三循环有向图，通过限制生成元组合至三个扇区，利用格点与镶嵌分析可达集增长，给出Moore型上界、显式构造及计算实验，揭示约束格点与极值问题的联系。
Automorphism groups of endomorphism monoids of $0$-unit $3$-valent circulant digraphs
本文研究 $0$-单位 $3$-度循环有向图的自同态幺半群的自同构群，通过建立无 $2$-圈单位循环有向图存在诱导有向圈的自同态的充要条件，显式确定了正规化子群 $U_S(\mathbb{Z}_n)$，解决了该族的未决问题。
A number field K is a finite extension of rational number field Q. A circulant digraph integral over K means that all its eigenvalues are algebraic integers of K. In this paper we give the sufficient and necessary condition for circulant digraphs which are integral over a number field K. And we solve the Conjecture3.3 in [XM] and find it is affirmative.
On 2-distance-transitive circulant digraphs
本文基于2-弧传递循环图的分类，完整分类了2-距离传递循环有向图，列出了所有同构类型，包括无向圈、完全多部图、Paley图及若干有向图族。
Let $D$ be a digraph. A subset $S$ of $V(D)$ is a stable set if every pair of vertices in $S$ is non-adjacent in $D$. A collection of disjoint paths $\mathcal{P}$ of $D$ is a path partition of $V(D)$, if every vertex in $V(D)$ is exactly on a path of $\mathcal{P}$. We say that a stable set $S$ and a path partition $\mathcal{P}$ are orthogonal if each path of $P$ contains exactly one vertex of $S$. A digraph $D$ satisfies the $α$-property if for every maximum stable set $S$ of $D$, there exists a path partition $\mathcal{P}$ such that $S$ and $\mathcal{P}$ are orthogonal. A digraph $D$ is $α$-d…
$3$-diregular circulant digraph on $12$ vertices is a counterexample of Jackson's conjecture about hamiltonicity of diregular digraphs
Oriented Trees in Digraphs without Oriented $4$-cycles
本文证明了在最小半度至少为k/2且不含4-圈的有向图中，只要最大出度和入度至少为k，就能包含任意具有k条弧的有向树，并对反有向树和树状结构给出更强结论。
Motivated by the Caccetta-Haggkvist Conjecture, we prove that every digraph on n vertices with minimum outdegree 0.3465n contains an oriented triangle. This improves the bound of 0.3532n of Hamburger, Haxell and Kostochka. The main new tool we use in our proof is the theory of flag algebras developed recently by Razborov.
In 2006, Noga Alon raised the following open problem: Does there exist an absolute constant $c>0$ such that every $2n$-vertex digraph with minimum out-degree at least $s$ contains an $n$-vertex subdigraph with minimum out-degree at least $\frac{s}{2}-c$ ? In this note, we answer this natural question in the negative, by showing that for arbitrarily large values of $n$ there exists a $2n$-vertex tournament with minimum out-degree $s=n-1$, in which every $n$-vertex subdigraph contains a vertex of out-degree at most $\frac{s}{2}-\left(\frac{1}{2}+o(1)\right)\log_3(s)$.
A study on Type-2 isomorphic circulant graphs. Part 7: Isomorphism series, digraph and graph of $C_n(R)$
定义循环图的同构集、同构系列、同构有向图与同构图，针对特定阶数循环图给出实例，并发现若干对循环图在模3下既非第一类也非第二类同构。
For nonnegative integers $k$ and $l$, let $\mathscr{D}(k,l)$ denote the family of digraphs in which every vertex has either indegree at most $k$ or outdegree at most $l$. In this paper we prove that the edges of every digraph in $\mathscr{D}(3,3)$ and $\mathscr{D}(4,4)$ can be covered by at most five directed cuts and present an example in $\mathscr{D}(3,3)$ showing that this result is best possible.
The minimum semi-degree of a digraph D is the minimum of its minimum outdegree and its minimum indegree. We show that every sufficiently large digraph D with minimum semi-degree at least n/2 +k-1 is k-linked. The bound on the minimum semi-degree is best possible and confirms a conjecture of Manoussakis from 1990. We also determine the smallest minimum semi-degree which ensures that a sufficiently large digraph D is k-ordered, i.e. that for every ordered sequence of k distinct vertices of D there is a directed cycle which encounters these vertices in this order.
Vertex-disjoint cycles of different lengths in tournaments
本研究围绕竞赛图中不同长度顶点不交圈的猜想展开，证明当k≥5时最小出度≥2k-1的竞赛图含k个顶点不交圈且其中3个长度不同，还证实最小出度≥6的竞赛图含3个不同长度的顶点不交圈且条件紧，解答了Chen与Chang的问题。
Let $n$ be sufficiently large and suppose that $G$ is a digraph on $n$ vertices where every vertex has in- and outdegree at least $n/2$. We show that $G$ contains every orientation of a Hamilton cycle except, possibly, the antidirected one. The antidirected case was settled by DeBiasio and Molla, where the threshold is $n/2+1$. Our result is best possible and improves on an approximate result by Häggkvist and Thomason.
Given a digraph D, the minimum semi-degree of D is the minimum of its minimum indegree and its minimum outdegree. D is k-ordered Hamiltonian if for every ordered sequence of k distinct vertices there is a directed Hamilton cycle which encounters these vertices in this order. Our main result is that every digraph D of sufficiently large order n with minimum semi-degree at least (n+k)/2 -1 is k-ordered Hamiltonian. The bound on the minimum semi-degree is best possible. An undirected version of this result was proved earlier by Kierstead, Sárközy and Selkow.
The Turan bound is a famous result in graph theory, which relates the independence number of an undirected graph to its edge density. Also the Caro-Wei inequality, which gives a more refined bound in terms of the vertex degree sequence of a graph, might be regarded today as a classical result. We show how these statements can be generalized to directed graphs, thus yielding a bound on directed feedback vertex number in terms of vertex outdegrees and in terms of average outdegree, respectively.
Weak and strong local irregularity of digraphs
针对有向图局部不规则性的研究空白，提出弱（基于入度-出度对）和强（基于平衡度）局部不规则性定义，建立分解与着色框架，结合有向图结构性质提出最小颜色数猜想并提供结果支持。
Path homology of circulant digraphs
本文通过位移自同构和傅里叶分解，提出了一种计算环状有向图路径同调的符号-矩阵方法，揭示了素数与合数n的依赖性及连接集S的稳定性现象。
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…
An edge-coloured graph $G$ is {\it rainbow connected} if any two vertices are connected by a path whose edges have distinct colours. This concept was introduced by Chartrand et al. in \cite{ch01}, and it was extended to oriented graphs by Dorbec et al. in \cite{DI}. In this paper we present some results regarding this extention, mostly for the case of circulant digraphs.
A digraph is 2-regular if every vertex has both indegree and outdegree two. We define an embedding of a 2-regular digraph to be a 2-cell embedding of the underlying graph in a closed surface with the added property that for every vertex~$v$, the two edges directed away from $v$ are not consecutive in the local rotation around $v$. In other words, at each vertex the incident edges are oriented in-out-in-out. The goal of this article is to provide an analogue of Whitney's theorem on planar embeddings in the setting of 2-regular digraphs. In the course of doing so, we note that Tutte's Theorem on…
The WL-rank of a digraph $Γ$ is defined to be the rank of the coherent configuration of $Γ$. The WL-dimension of $Γ$ is defined to be the smallest positive integer $m$ for which $Γ$ is identified by the $m$-dimensional Weisfeiler-Leman algorithm. We classify the Deza circulant graphs of WL-rank $4$. In additional, it is proved that each of these graphs has WL-dimension at most $3$. Finally, we establish that some families of Deza circulant graphs have WL-rank $5$ or $6$ and WL-dimension at most $3$.
Cyclic structure, vertex degree and number of linear vertices in Minimal Strong Digraphs
本研究针对极小强有向图（MSDs），推导了线性顶点数与顶点度的关系、最长环长度的上界，拓展了其特征多项式系数的界，并证明MSD最长环计算问题为NP-Hard。
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。
The paper is concerned with directed versions of Posa's theorem and Chvatal's theorem on Hamilton cycles in graphs. We show that for each a>0, every digraph G of sufficiently large order n whose outdegree and indegree sequences d_1^+ \leq ... \leq d_n^+ and d_1^- \leq >... \leq d_n^- satisfy d_i^+, d_i^- \geq min{i + a n, n/2} is Hamiltonian. In fact, we can weaken these assumptions to (i) d_i^+ \geq min{i + a n, n/2} or d^-_{n - i - a n} \geq n-i; (ii) d_i^- \geq min{i + a n, n/2} or d^+_{n - i - a n} \geq n-i; and still deduce that G is Hamiltonian. This provides an approximate version of a …
Let $G$ be a digraph with adjacency matrix $A(G)$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. Nikiforov \cite{Niki} proposed to study the convex combinations of the adjacency matrix and diagonal matrix of the degrees of undirected graphs. Liu et al. \cite{LWCL} extended the definition to digraphs. For any real $α\in[0,1]$, the matrix $A_α(G)$ of a digraph $G$ is defined as $$A_α(G)=αD(G)+(1-α)A(G).$$ The largest modulus of the eigenvalues of $A_α(G)$ is called the $A_α$ spectral radius of $G$, denoted by $λ_α(G)$. This paper proves some extremal results about the spe…
New classes of groups related to algebraic combinatorics with applications to isomorphism problems
本文引入$5/2$-群类的两个精化，证明其正则循环子群共轭，从而推广并简化了对称构型与单位循环有向图同构的判定证明。
Periodic orbits on 2-regular circulant digraphs
本文研究了2-正则Cayley有向图上的周期轨道，用步数确定了给定长度的本原周期轨道数量，描述了其格结构，并通过环绕数表征重复数，同时计算了双字母字母表上的Lyndon字数量。
We give a quasi-complete solution of the (Δ,N) problem for two well-known families of digraphs used as good models for large interconnection networks. In our study we also relate both families, the New Amsterdam and Manhattan digraphs, with the double-step graphs (or circulant graphs with degree two).
We say that a (di)graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. The seminal Hajnal--Szemerédi theorem characterises the minimum degree that ensures a graph $G$ contains a perfect $K_r$-packing. In this paper we prove the following analogue for directed graphs: Suppose that $T$ is a tournament on $r$ vertices and $G$ is a digraph of sufficiently large order $n$ where $r$ divides $n$. If $G$ has minimum in- and outdegree at least $ (1-1/r)n$ then $G$ contains a perfect $T$-packing. In the case when $T$ is a cyclic …
An edge-coloured path is \emph{rainbow} if its edges have distinct colours. An edge-coloured connected graph is said to be \emph{rainbow connected} if any two vertices are connected by a rainbow path, and \emph{strongly rainbow connected} if any two vertices are connected by a rainbow geodesic. The (\emph{strong}) \emph{rainbow connection number} of a connected graph is the minimum number of colours needed to make the graph (strongly) rainbow connected. These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. As an extension, Krivelevich and Yuster proposed the…
Antidirected forests in digraphs
本文证明每个弧数超过阈值 $g_k(n)$ 的有向图必包含任意给定 $k$ 条弧的反定向森林，并确定最大有向极值数为 $2\mathrm{ex}(n,kK_2)$，方法结合计数不等式、嵌入与分解。
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。
In a directed multigraph, the imbalance of a vertex $v_{i}$ is defined as $b_{v_{i}}=d_{v_{i}}^{+}-d_{v_{i}}^{-}$, where $d_{v_{i}}^{+}$ and $d_{v_{i}}^{-}$ denote the outdegree and indegree respectively of $v_{i}$. We characterize imbalances in directed multigraphs and obtain lower and upper bounds on imbalances in such digraphs. Also, we show the existence of a directed multigraph with a given imbalance set.
Variation in $α$ trace norm of a digraph by deletion of a vertex or an arc and its applications
本文研究了有向图在删除顶点或弧后其 $α$ 迹范数的变化规律，并据此刻画了具有最大 $α$ 迹范数的定向树与单圈有向图。
The family $D(k,m)$ of graphs having an orientation such that for every vertex $v \in V(G)$ either (outdegree) $°^+(v) \le k$ or (indegree) $°^-(v) \le m$ have been investigated recently in several papers because of the role $D(k,m)$ plays in the efforts to estimate the maximum directed cut in digraphs and the minimum cover of digraphs by directed cuts. Results concerning the chromatic number of graphs in the family $D(k,m)$ have been obtained via the notion of $d$-degeneracy of graphs. In this paper we consider a far reaching generalization of the family $D(k,m)$, in a complementary form, int…
Towards inductive proofs in algebraic combinatorics
本文引入了一类新的传递置换群，给出了其商保持在该族中的充分条件，并利用归纳法确定了阶为$p^n$的循环有向图的自同构群。
Structure of cycles in Minimal Strong Digraphs
本文研究了极小强有向图（MSD）中圈的结构，通过抑制圈弧并收缩强连通分量得到Hasse图，证明了线性顶点的多个存在性定理及其数量下界。
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。
Locating a Seymour vertex in oriented graphs having minimum outdegree at most 7
通过局部约简和OR-Tools CP-SAT不可行性检验，证明了最小出度为7的有向图的Seymour第二邻域猜想，将阈值从6提升至7。
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