Louis Esperet, Laetitia Lemoine, Frédéric Maffray
Let $G$ be a directed planar graph on $n$ vertices, with no directed cycle of length less than $g\ge 4$. We prove that $G$ contains a set $X$ of vertices such that $G-X$ has no directed cycle, and $|X|\le \tfrac{5n-5}9$ if $g=4$, $|X|\le \tfrac{2n-5}4$ if $g=5$, and $|X|\le \tfrac{2n-6}{g}$ if $g\ge 6$. This improves recent results of Golowich and Rolnick.
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Planar digraphs of digirth five are 2-colorable
Neumann-Lara（1985）和Škrekovski证明了任意数位数至少为3的平面有向图是2-可着色的。我们证明了这个猜想的一个放松的版本：任何数位数至少为5的平面有向图是2-可着色的。结果也适用于列表着色的设置。
We study vertex-quasiprimitive $2$-arc-transitive digraphs, reducing the problem of vertex-primitive $2$-arc-transitive digraphs to almost simple groups. This includes a complete classification of vertex-quasiprimitive $2$-arc-transitive digraphs where the action on vertices has O'Nan-Scott type SD or CD.
We solve the long-standing existence problem of vertex-primitive 2-arc-transitive digraphs by constructing an infinite family of such digraphs.
We study $G$-vertex-primitive and $(G,s)$-arc-transitive digraphs for almost simple groups $G$ with socle $\mathrm{PSL}_n(q)$. It turns out that $s\leqslant2$ for such digraphs, which provides the first step in determining an upper bound on $s$ for all the vertex-primitive $s$-arc-transitive digraphs.
In 2017, Giudici, Li and the third author constructed the first known family of vertex-primitive $2$-arc-transitive digraphs of valency at least $2$. The smallest digraph in this family admits $\mathrm{PSL}_3(49)$ acting $2$-arc-transitively with vertex-stabilizer $\mathrm{A}_6$ and hence has $30758154560$ vertices. In this paper, we prove that this digraph is the vertex-primitive $2$-arc-transitive digraph of valency at least $2$ with fewest vertices.
A rooted digraph is a vertex-flame if for every vertex $v$ there is a set of internally disjoint directed paths from the root to $v$ whose set of terminal edges covers all ingoing edges of $v$. It was shown by Lovász that every finite rooted digraph admits a spanning subdigraph which is a vertex-flame and large, where the latter means that it preserves the local connectivity to each vertex from the root. A structural generalisation of vertex-flames and largeness to infinite digraphs was given by the third author and the analogue of Lovász' result for countable digraphs was shown. We strengthen…
Line digraphs and coreflexive vertex sets
引入上自反集的概念来研究有向图的结构。给出了线有向图和n阶线有向图的新刻画。余自反集也导致了另一种从给定的有向图形成交有向图的自然方法。
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$(1 - 3/g)n$。
On vosperian and superconnected vertex-transitive digraphs
研究了具有传递自同构群的有向图的结构，其中每个最小基数割集由某个顶点的所有后继者或所有前驱者组成。我们改进了这一领域的大部分现有结果。
A vertex coloring of a strong digraph $D$ is a \emph{strong vertex-monochromatic connection coloring (SVMC-coloring)} if for every pair $u, v$ of vertices in $D$ there exists an $(u,v)$-path having all its internal vertices of the same color. Let $smc_v(D)$ denote the maximum number of colors used in an SVMC-coloring of a digraph $D$. In this paper we determine the value of $smc_v(D)$, whenever $D$ is the line digraph of a digraph. Also, if $T$ is a tournament, we give conditions to find the exact value of $smc_v(T)$.
Let $D=(V,A)$ be a digraphs without isolated vertices. A vertex-degree based invariant $I(D)$ related to a real function $φ$ of $D$ is defined as a summation over all arcs, $I(D) = \frac{1}{2}\sum_{uv\in A}{φ(d_u^+,d_v^-)}$, where $d_u^+$ (resp. $d_u^-$) denotes the out-degree (resp. in-degree) of a vertex $u$. In this paper, we give the extremal values and extremal digraphs of $I(D)$ over all digraphs with $n$ non-isolated vertices. Applying these results, we obtain the extremal values of some vertex-degree based topological indices of digraphs, such as the Randić index, the Zagreb index, the…
For a fixed simple digraph $F$ and a given simple digraph $D$, an $F$-free $k$-coloring of $D$ is a vertex-coloring in which no induced copy of $F$ in $D$ is monochromatic. We study the complexity of deciding for fixed $F$ and $k$ whether a given simple digraph admits an $F$-free $k$-coloring. Our main focus is on the restriction of the problem to planar input digraphs, where it is only interesting to study the cases $k \in \{2,3\}$. From known results it follows that for every fixed digraph $F$ whose underlying graph is not a forest, every planar digraph $D$ admits an $F$-free $2$-coloring, a…
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…
On non-Hamiltonian circulant digraphs of outdegree three
我们构造了无穷多个出度为3的连通循环有向图，这些图没有哈密顿回路。我们所有的例子都有偶数个顶点，我们的例子有两种类型：或者有向图中的每个顶点都与两个直径相对的顶点相邻，或者每个顶点都与与它自己直径相对的顶点相邻。
Notes on the connectivity of Cayley coset digraphs
将Hamidoune关于分层Cayley有向图的连通性结果推广到Cayley陪集有向图，从而推广到任意点传递有向图。证明了如果Cayley陪集有向图可以按某种方式分层分解，则它是最优点连通的。结果是通过扩展Hamidoune使用的方法得到的。它们用来证明圈前缀图是最优点连通的。这意味着圈前缀图具有良好的容错特性。
We construct for all $ k\in \mathbb{N} $ a $ k $-edge-connected digraph $ D $ with $ s,t\in V(D) $ such that there are no edge-disjoint $ s \rightarrow t $ and $t\rightarrow s $ paths. We use in our construction "self-similar" graphs which technique could be useful in other problems as well.
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…
Quasiprimitive and bi-quasiprimitive highly-arc-transitive digraphs and finite simple groups
本文研究了准素性和双准素性高度弧传递有向图与有限单群的关系，提出新的构造方法并揭示了相关开放问题。
A digraph $D=(V(D)$, $A(D))$ of order $n\geq 3$ is pancyclic, whenever $D$ contains a directed cycle of length $k$ for each $k\in \{3,\ldots,n\}$; and $D$ is vertex-pancyclic iff, for each vertex $v\in V(D)$ and each $k\in \{3,\ldots,n\}$, $D$ contains a directed cycle of length $k$ passing through $v$. Let $D_1, D_2, \ldots, D_k$ be a collection of pairwise vertex disjoint digraphs. The generalized sum (g.s.) of $D_1, D_2, \ldots, D_k$, denoted by $\oplus_{i=1}^k D_i$ or $D_1\oplus D_2 \oplus \cdots \oplus D_k$, is the set of all digraphs $D$ satisfying: (i) $V(D)=\bigcup_{i=1}^k V(D_i)$, (ii…
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。
In the mid-1990s, two groups of authors independently obtained classifications of vertex-transitive graphs whose order is a product of two distinct primes. In the intervening years it has become clear that there is additional information concerning these graphs that would be useful, as well as making explicit the extensions of these results to digraphs. Additionally, there are several small errors in some of the papers that were involved in this classification. The purpose of this paper is to fill in the missing information as well as correct all known errors.
We show that every k-dichromatic vertex-critical digraph on at most 2k-2 vertices has a disconnected complement. This answers a question of Bang-Jensen et al., and generalises a classical theorem of Gallai on undirected vertex-critical graphs.
Neumann-Lara conjectured in 1985 that every planar digraph with digirth at least three is 2-colourable, meaning that the vertices can be 2-coloured without creating any monochromatic directed cycles. We prove a relaxed version of this conjecture: every planar digraph of digirth at least four is 2-colourable.
The Second Neighborhood Conjecture states that every simple digraph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle intersection graph of an even graph is a new graph whose vertices are the cycles in a cycle decomposition of the original graph and whose edges represent vertex intersections of the cycles. By using a digraph variant of this concept, we prove that Eulerian digraphs which admit a simple cycle intersection graph have not only adhere to the Second Neighborhood Conjecture, but …
The minimal nonplanar strong digraphs
本文将 Kuratowski 定理推广到有向图，刻画了在子有向图包含关系下最小的强连通非平面、非外平面及非串并联有向图。
A conjecture by Lichiardopol states that for every $k \ge 1$ there exists an integer $g(k)$ such that every digraph of minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. Motivated by Lichiardopol's conjecture, we study the existence of vertex-disjoint directed cycles satisfying length constraints in digraphs of large connectivity or large minimum degree. Our main result is that for every $k \in \mathbb{N}$, there exists $s(k) \in \mathbb{N}$ such that every strongly $s(k)$-connected digraph contains $k$ vertex-disjoint directed cycles …
Long Directed Cycles in Vertex-Transitive Digraphs
在图论与群论交叉领域，寻找顶点传递图和有向图中的哈密顿圈是经典问题。本文构造无限多个最长有向圈遗漏至少\(n/12\)个顶点的连通顶点传递有向图，证实相关猜想，还将有向圈长度下限从\(\Omega(n^{1/3})\)提高到\(\Omega(\sqrt n)\) 。
Bounding $s$ for vertex-primitive $s$-arc-transitive digraphs of alternating and symmetric groups
本文针对 Praeger 提出的有限顶点本原 s-弧传递有向图的上界问题，在群为交错群或对称群的情形下证明了 s ≤ 2。
Arc-connectivity and super arc-connectivity of mixed Cayley digraph
有向图X=（V，E）是max-λ，如果λ（X）=δ（X）.一个有向图X是超λ的，如果X的每一个最小割是某个顶点的内弧集或某个顶点的外弧集。本文证明了除了少数例外，强连通混合Cayley有向图都是max-λ和super-λ图。
Spanning Factorizations in Vertex-Transitive Digraphs of Degree 2
针对出度为d的顶点传递有向图中生成1-因子分解的存在性问题，本文聚焦d=2的情况，利用交替环和块系结构提出块/阶段框架，得到同时包含两条出边的充分条件，证明块障碍的影响及强传递集的存在性，并给出可行块大小约束等结果。
On the 2-Linkage Problem for Split Digraphs
本文证明了6强分裂有向图和5强半完全分裂有向图均为2-连通，解决了相关问题。
It follows from a theorem of Lovász that if $ D $ is a finite digraph with $ r\in V(D) $ then there is a spanning subdigraph $ E $ of $ D $ such that for every vertex $ v\neq r $ the following quantities are equal: the local connectivity from $ r $ to $ v $ in $ D $, the local connectivity from $ r $ to $ v $ in $ E $ and the indegree of $ v $ in $ E $. In infinite combinatorics cardinality is often an overly rough measure to obtain deep results and it is more fruitful to capture structural properties instead of just equalities between certain quantities. The best known example for such a resu…
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 …
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 $k\le 2$.
Vertex-disjoint directed and undirected cycles in general digraphs
有向图D的双截线数t（D）是D的双截线的最小尺寸i。e. D的顶点的集合T，使得D-T是非循环的。本文研究如下问题：给定一个有向图D，判定D中是否存在一个有向图B，D的基础无向图中是否存在一个圈C，使得B，C不相交。众所周知，当限制到强连通图时，对于这个问题存在一个多项式时间算法，如果B，C存在，它实际上会找到它们。我们把这个问题推广到任何一类有向图D，其中t（D）不等于1或t（D）=1，并且有界的双截线数，并证明了对于一类特殊的有向图D，其中t（D）=1，这个问题是NP-完全的，因此，在一般情况下。
The energy and iota energy of signed digraphs are respectively defined as the sum of absolute values of real parts and sum of absolute values of imaginary parts of its eigenvalues. Recently, Yang and Wang (2018) find the energy and iota energy ordering of digraphs in D(n) and compute the maximal energy and iota energy, where D(n) denotes the set of vertex-disjoint bicyclic digraphs of a fixed order n. In this paper, we investigate the energy ordering of signed digraphs in D(n,s) and finds the maximal energy, where D(n,s) denotes the set of vertex-disjoint bicyclic sidigraphs of a fixed order n…
Switching Reconstruction of Digraphs
有向图中关于顶点的切换意味着反转与该顶点关联的每条边的方向。Bondy和Mercier提出了一个问题，即一个有向图是否可以从有向图的同构类型的多个集合中重构到同构，这些同构类型是通过关于每个顶点的交换得到的。由于已知的最大的不可重构的定向图有8个顶点，这是很自然的问是否有任何更大的不可重构的图。在本文中，我们继续调查这个问题。我们发现，有44个不可重构的定向图图的底层无向图的最大度不超过2。我们还确定了开关稳定的定向图，这是那些图的所有开关返回一个有向图同构到原来的完整集。
We prove that every digraph has a vertex 4-colouring such that for each vertex $v$, at most half the out-neighbours of $v$ receive the same colour as $v$. We then obtain several results related to the conjecture obtained by replacing 4 by 3.
In this paper, we determine the maximum size of digraphs on $n$ vertices in which there are no two distinct walks of length $3$ with the same initial vertex and the same terminal vertex. The digraphs attaining this maximum size are also characterized. Combining this with previous results, we obtain a full solution to a problem proposed by X. Zhan in 2007.
Vertex-primitive digraphs having vertices with almost equal neighbourhoods
我们考虑两个顶点具有几乎相等邻域的顶点本原有向图（即，一个顶点的邻域而不是另一个顶点的邻域的顶点集很小）。我们证明了这样的有向图的结构结果，然后将其应用于回答一个问题的Araújo和卡梅隆同步组。
Digraph Yama Nim
本文研究两堆Nim变体Yama Nim的获胜策略与SG值，提出其推广形式Digraph Yama Nim，证明该推广游戏的胜者判定问题即使在二分有向无环图下仍是PSPACE完全的，同时给出了部分可解情形。
The Number of Seymour Vertices in Random Tournaments and Digraphs
西摩距离2猜想指出，在任何有向图中，都存在一个顶点（“西摩顶点”），它在距离2处的邻居至少和在距离1处的邻居一样多。我们探索的有效性的概率语句沿线建议西摩的猜想，证明几乎肯定有一个“大”的西摩顶点的随机比赛和“甚至更多”一般随机有向图。
Feedback vertex sets of digraphs with bounded maximum degree
该研究在最大度有界的情况下，给出了有向图反馈顶点集问题的最优上界，针对不同图结构给出了不同的界值。
Let $H = (V_H, A_H)$ be a digraph which may contain loops, and let $D = (V_D, A_D)$ be a loopless digraph with a coloring of its arcs $c: A_D \to V_H$. An $H$-walk of $D$ is a walk $(v_0, \dots, v_n)$ of $D$ such that $(c(v_{i-1}, v_i), c(v_i, v_{i+1}))$ is an arc of $H$, for every $1 \le i \le n-1$. For $u, v \in V_D$, we say that $u$ reaches $v$ by $H$-walks if there exists an $H$-walk from $u$ to $v$ in $D$. A subset $S \subseteq V_D$ is a kernel by $H$-walks of $D$ if every vertex in $V_D \setminus S$ reaches by $H$-walks some vertex in $S$, and no vertex in $S$ can reach another vertex in…
Point- and arc-reaching sets of vertices in a digraph
在有向图$D = (X, \mathcal{U})$中，不一定是有限的，如果存在源自$u$并包含$(x, y)$的有向行走$W$，则弧$(x, y) \in \mathcal{U}$是从顶点$u$可达的。子集$S \subseteq X$是$D$的弧到达集合，如果对于每个弧$(x, y)$，存在起源于顶点$u \in S$并且包含$(x, y)$的双行走$W$。最小弧可达集是弧基。$S$是一个可达点集，如果对于每个顶点$v$，都存在一个从$W$到$v$的双向行走，该双向行走起源于顶点$u \in S$。最小可达点集是点基。我们将Harary，Norman和Cartwright关于有限有向图中点基的结果推广到无限有向图中的点基和弧基。
On the dib-chromatic number of a digraph
本文针对有向图的dib-色数展开研究，明确了其存在性，并对二部有向图的dib-色数进行了分析，该色数是满足特定顶点指向条件的无环着色的最大颜色数。
A subset $S$ of vertices of a digraph $D$ is a double dominating set (total $2$-dominating set) if every vertex not in $S$ is adjacent from at least two vertices in $S$, and every vertex in $S$ is adjacent from at least one vertex in $S$ (the subdigraph induced by $S$ has no isolated vertices). The double domination number (total $2$-domination number) of a digraph $D$ is the minimum cardinality of a double dominating set (total $2$-dominating set) in $D$. In this work, we investigate these concepts which can be considered as two extensions of double domination in graphs to digraphs, along wit…
Maximally connected and super arc-connected Bi-Cayley digraphs
设X=（V，E）是一个有向图. X是极大连通的，如果κ（X）=δ（X）.若λ（X）=δ（X），则X是极大弧连通的. X是超弧连通的，如果X的每个最小弧割是某个顶点的内弧集或某个顶点的外弧集。本文证明了强连通的有向Bi-Cayley图是极大连通的和极大弧连通的，并且大多数强连通的有向Bi-Cayley图是超弧连通的.
Long cycles in vertex transitive digraphs
研究了顶点传递有向图中长环的存在性，证明了在n个顶点的图中可找到长度至少为Ω(n^{1/3})的有向环。
Perfect codes in 2-valent Cayley digraphs on abelian groups
研究阿贝尔群上强连通2价Cayley有向图的完美码问题，完成图的分类并完全确定其所有完美码。
A fascinating problem on digraphs is the existence problem of the finiteupper bound on s for all vertex-primitive s-arc-transitive digraphs except directed cycles (which is known to be reduced to the almost simple groups case). In this paper,we prove that s 2 for all G-vertex-primitive s-arc-transitive digraphs with G an (insoluble) alternating or symmetric group, which makes an important progress towards a solution of the problem. The proofs involves some methods that may be used to investigate other almost simple groups cases.
Vertex-primitive digraphs with large fixity
本文定义了有向图的相对固定性，刻画了相对固定性至少为 $1/3$ 的顶点本原有向图，并证明了有界出度下仅有有限多个此类图。
Remoteness, order, size and connectivity constraints in digraphs
本文针对强连通有向图，给出了在给定阶、大小和顶点连通性条件下remoteness的尖锐上界，刻画了极值图，并将相关上界推广至欧拉有向图。
On the Complexity of Digraph Colourings and Vertex Arboricity
本文研究有向图与图的分数量着色相关决策问题的复杂性，证明星色数、分数量色数及循环顶点树荫度的判定问题在特定条件下均为NP完全，并引入循环同态概念以证明关键结果。
For a positive integer $k$, a $k$-rainbow dominating function ($k$RDF) on a digraph $D$ is a function $f$ from the vertex set $V(D)$ to the set of all subsets of $\{1,2,\ldots,k\}$ such that for any vertex $v$ with $f(v)=\emptyset$, $\bigcup_{u\in N^-(v)}f(u)=\{1,2,\ldots,k\}$, where $N^-(v)$ is the set of in-neighbors of $v$. The weight of a $k$RDF $f$ is defined as $\sum_{v\in V(D)}|f(v)|$. A $k$RDF $f$ on $D$ with no isolated vertex is called a total $k$-rainbow dominating function if the subdigraph of $D$ induced by the set $\{v\in V(D):f(v)\ne\emptyset\}$ has no isolated vertex. The total…
Matrices with simple symmetric digraphs and their group inverses
本文定义了简单对称有向图类 $\mathcal{D}$，给出了其群逆元素的图论描述，并对实方阵 $A$ 及其群逆 $A^{\\#}$ 均属于该类的矩阵进行了分类。
When all directed cycles have the same weight
本文研究了所有有向环具有相同权重的有向图结构，提出两种构造方法并设计多项式时间算法来判断有向图是否可加权。
A set $\mathcal{S}$ of derangements (fixed-point-free permutations) of a set $V$ generates a digraph with vertex set $V$ and arcs $(x,x^σ)$ for $x\in V$ and $σ\in\mathcal{S}$. We address the problem of characterising those infinite (simple loopless) digraphs which are generated by finite sets of derangements. The case of finite digraphs was addressed in earlier work by the second and third authors. A criterion is given for derangement generation which resembles the criterion given by De Bruijn and Erdős for vertex colourings of graphs in that the property for an infinite digraph is determined …
A note on heavy cycles in weighted digraphs
加权有向图是指每个弧都被赋予一个非负数的有向图，称为弧的权。加权有向图$D$中顶点$v$的加权出度是以$v$为尾的弧的权之和，$D$中有向圈$C$的权是$C$中弧的权之和.本文证明了：如果一个阶为$n$的加权有向图$D$的每个顶点的加权出度至少为1，则$D$中存在一个权至少为$1/\log_2 n$的有向圈.这证明了一个猜想Bollobás和斯科特了一个常数的因素。
A Remark on the Second Neighborhood Problem
Seymour的第二邻域猜想指出，每个简单有向图（没有对角）都有一个顶点，其第一个外邻域至多与第二个外邻域一样大。这样的顶点被称为具有第二邻域属性（SNP）。我们定义了“好”的有向图，并证明了一个声明，这意味着每个饲料顶点的比赛有SNP。在有向图缺少匹配的情况下，我们通过改进Fidler和Yuster的证明并使用良好的有向图来展示具有SNP的饲料顶点。此外，在某些情况下，我们展示了两个顶点与SNP。
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