István Kovács, Mikhail Muzychuk, Péter P. Pálfy, Grigory Ryabov, Gábor Somlai
We prove that the direct product of two coprime order elementary abelian groups of rank two, as well as the direct product of a cyclic group of prime order and a cyclic group of square free order are DCI-groups. The latter is a generalization of Muzychuk's result on cyclic groups (J. Combin. Theory Ser. A, 1995).
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On CI-property of normal Cayley digraphs over abelian groups
本文研究阿贝尔群上正规 Cayley 有向图的 CI-性质，将任意阿贝尔群情形归约为阿贝尔 p-群，并证明了阶数不超过 $p^5$ 的阿贝尔 p-群上的正规 Cayley 有向图均为 CI 的。
We investigate the class of infinite distance-transitive digraphs $D$ of finite out-valency. We show that if $D$ is a weakly descendant-homogeneous in such a class then either (1) $D$ has property $Z$ and the reachability relation is not universal; or (2) $D$ does not have property $Z$, the reachability relation is universal and $D$ has infinite in-valency. Also, we show that earlier results, proved in the context of highly-arc-transitive digraphs, hold under the weaker condition of distance-transitivity. Finally, we give a description of a subclass of the class distance-transitive weakly desc…
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 …
Let $D$ be a digraph on $p\geq 5$ vertices with minimum degree at least $p-1$ and with minimum semi-degree at least $p/2-1$. For $D$ (unless some extremal cases) we present a detailed proof of the following results [12]: (i) $D$ contains cycles of length 3, 4 and $p-1$; (ii) if $p=2n$, then $D$ is hamiltonian.
A classical result by Erdos and Posa states that there is a function $f: {\mathbb N} \rightarrow {\mathbb N}$ such that for every $k$, every graph $G$ contains $k$ pairwise vertex disjoint cycles or a set $T$ of at most $f(k)$ vertices such that $G-T$ is acyclic. The generalisation of this result to directed graphs is known as Younger's conjecture and was proved by Reed, Robertson, Seymour and Thomas in 1996. This so-called Erdos-Posa-property can naturally be generalised to arbitrary graphs and digraphs. Robertson and Seymour proved that a graph $H$ has the Erdos-Posa-property if, and only if…
A polytopal digraph $G(P)$ is an orientation of the skeleton of a convex polytope $P$. The possible non-degenerate pivot operations of the simplex method in solving a linear program over $P$ can be represented as a special polytopal digraph known as an LP digraph. Presently there is no general characterization of which polytopal digraphs are LP digraphs, although four necessary properties are known: acyclicity, unique sink orientation(USO), the Holt-Klee property and the shelling property. The shelling property was introduced by Avis and Moriyama (2009), where two examples are given in $d=4$ d…
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…
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to $S$ is said to be normal if the right regular representation of $G$ is normal in the automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if there is $α\in Aut(G)$ such that $S^α=T$, whenever $Cay(G,S)\cong Cay(G,T)$ for a Cayley (di)graph $Cay(G,T)$. A finite group $G$ is called a DCI-group or a NDCI-group if all Cayley digraphs or normal Cayley digraphs of $G$ are CI-digraphs, and is called a CI-group or a NCI-group if all Cayley graphs or normal Cayley graphs of $G$ are CI-graphs, respectively. Motivated by a conjecture …
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called normal if the right regular representation of $G$ is a normal subgroup in the full automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if for every $T\subseteq G$, $Cay(G,S)\cong Cay(G,T)$ implies that there is $σ\in Aut(G)$ such that $S^σ=T$. We call a group $G$ a NDCI-group if all normal Cayley digraphs of $G$ are CI-digraphs, and a NCI-group if all normal Cayley graphs of $G$ are CI-graphs, respectively. In this paper, we prove that a cyclic group of order $n$ is a NDCI-group if and only if…
A Cayley digraph $\rm{Cay}(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called a CI-digraph if for any Cayley digraph $\rm{Cay}(G,T)$ isomorphic to $\rm{Cay}(G,S)$, there is an $α\in \rm{Aut}(G)$ such that $S^α=T$. For a positive integer $m$, $G$ is said to have the $m$-DCI property if all Cayley digraphs of $G$ with out-valency $m$ are CI-digraphs. Li [The Cyclic groups with the $m$-DCI Property, European J. Combin. 18 (1997) 655-665] characterized cyclic groups with the $m$-DCI property, and in this paper, we characterize dihedral groups with the $m$-DCI property. For a dihedr…
We introduce adequate concepts of expansion of a digraph to obtain a sequential construction of minimal strong digraphs. We characterize the class of minimal strong digraphs whose expansion preserves the property of minimality. We prove that every minimal strong digraph of order $n\geq 2$ is the expansion of a minimal strong digraph of order $n-1$ and we give sequentially generative procedures for the constructive characterization of the classes of minimal strong digraphs. Finally we describe algorithms to compute unlabeled minimal strong digraphs and their isospectral classes.
A majority digraph is a finite simple digraph $G=(V,\to)$ such that there exist finite sets $A_v$ for the vertices $v\in V$ with the following property: $u\to v$ if and only if "more than half of the $A_u$ are $A_v$". That is, $u\to v$ if and only if $ |A_u \cap A_v | > \frac{1}{2} \cdot |A_u|$. We characterize the majority digraphs as the digraphs with the property that every directed cycle has a reversal. If we change $\frac{1}{2}$ to any real number $α\in (0,1)$, we obtain the same class of digraphs. We apply the characterization result to obtain a result on the logic of assertions "most $X…
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 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 $α$-diperfect…
A detailed description of the structure of two-ended arc-transitive digraphs is given. It is also shown that several sets of conditions, involving such concepts as Property Z, local quasi-primitivity and prime out-valency, imply that an arc-transitive digraph must be highly-arc-transitive. These are then applied to give a complete classification of two-ended highly-arc-transitive digraphs with prime in- and out-valencies.
Generalized quaternion groups with the m-DCI property
本文研究广义四元数群Q_{4n}的m-DCI性质，证明了当1≤m≤2n-1时n必为奇数且不被任何不超过m-1的素数的平方整除，并给出n为素数幂时的完整刻画。
Enumerating Cayley digraphs on dihedral groups
该研究聚焦二面体群上Cayley有向图的CI-同构计数，借助Cauchy-Frobenius引理与自同构性质，推导了具有DCI性质的二面体群（特别是$\text{D}_{6p}$，p为大于3的素数）上非同构Cayley有向图数的显式公式。
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。
Cyclic Neighborhoods in Digraphs
针对满足任意顶点邻域非循环的有向图，证明强连通时边数下界为7n/3、强2-连通时为8n/3，两个下界均为最优。
We extend the Grundy number and the ochromatic number, parameters on graph colorings, to digraph colorings, we call them {\emph{digrundy number}} and {\emph{diochromatic number}}, respectively. First, we prove that for every digraph the diochromatic number equals the digrundy number (as it happen for graphs). Then, we prove the interpolation property and the Nordhaus-Gaddum relations for the digrundy number, and improve the Nordhaus-Gaddum relations for the dichromatic and diachromatic numbers bounded previously by the authors in [Electron. J. Combin. 25 (2018) no. 3, Paper {\#} 3.51, 17 pp.]
In this paper we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group $R$ has a nontrivial proper n…
A Cayley graph Cay$(G;S)$ has the CI (Cayley Isomorphism) property if for every isomorphic graph Cay$(G;T)$, there is a group automorphism $α$ of $G$ such that $S^α=T$. The DCI (Directed Cayley Isomorphism) property is defined analogously on digraphs. A group $G$ is a CI-group if every Cayley graph on $G$ has the CI property, and is a DCI-group if every Cayley digraph on $G$ has the DCI property. Since a graph is a special type of digraph, this means that every DCI-group is a CI-group, and if a group is not a CI-group then it is not a DCI-group, but there are well-known examples of groups that…
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…
A finite group $G$ is called a DCI-group if two Cayley digraphs over $G$ are isomorphic if and only if their connection sets are conjugate by a group automorphism. We prove that the group $C_4\times C_p^2$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$.
Partitions of an Eulerian Digraph into Circuits
本文研究连通欧拉有向图的消去性质，通过碎片堆理论给出替代证明，将马丁多项式与交图色多项式关联，并应用于推导Harary-Sachs定理。
Gordon and McMahon defined a two-variable greedoid polynomial $ f(G;t,z) $ for any greedoid $ G $. They studied greedoid polynomials for greedoids associated with rooted graphs and rooted digraphs. They proved that greedoid polynomials of rooted digraphs have the multiplicative direct sum property. In addition, these polynomials are divisible by $ 1 + z $ under certain conditions. We compute the greedoid polynomials for all rooted digraphs up to order six. A greedoid polynomial $ f(D) $ of a rooted digraph $ D $ of order $ n $ GM-factorises if $ f(D) = f(G) \cdot f(H) $ such that $ G $ and $ H…
On isomorphisms of $m$-Cayley digraphs
本文研究m-Cayley有向图的同构问题，通过刻画自同构群中G的正规化子并推广CI性质，定义了mCI、mPCI、mDCI和mPDCI等概念，给出Babai型判据，并分类了m≥2时的有限mDCI群和m≥4时的有限mPDCI群。
We consider random directed graphs, and calculate the distribution of the cokernels of their laplacian, following the methods used by Wood. As a corollary, we show that the probability that a random digraph is coeulerian is asymptotically upper bounded by a constant around $0.43$.
We define $k$-dimensional digraphs and initiate a study of their spectral theory. The $k$-dimensional digraphs can be viewed as generating graphs for small categories called $k$-graphs. Guided by geometric insight, we obtain several new series of $k$-graphs using cube complexes covered by Cartesian products of trees, for $k \geq 2$. These $k$-graphs can not be presented as virtual products, and constitute novel models of such small categories. The constructions yield rank-$k$ Cuntz-Krieger algebras for all $k\geq 2$. We introduce Ramanujan $k$-graphs satisfying optimal spectral gap property, a…
The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group $G$ over a local field $F$. We show that if $T$ is any $k$-regular $G$-equivariant operator on the Bruhat-Tits building with a simple combinatorial property (collision-free), the associated random walk on the $n$-vertex Ramanujan complex has cutoff at time $\log_k n$. The high dimensional case, unlike that of graphs, requires tools from non-commutat…
On the automorphism group of direct product of digraphs
研究有向图直积自同构群等于因子自同构群直积的条件，给出连通有向图不稳定的充要条件，并应用于循环有向图。
On Isomorphisms of Tetravalent Cayley Digraphs over Dihedral Groups
本文研究二面体群上四度Cayley有向图的同构，给出了4-DCI-群和4-CI-群的充要条件，即n为奇数且不被9整除时是4-DCI-群，n为奇数时是4-CI-群。
A dominating (respectively, total dominating) set $S$ of a digraph $D$ is a set of vertices in $D$ such that the union of the closed (respectively, open) out-neighborhoods of vertices in $S$ equals the vertex set of $D$. The minimum size of a dominating (respectively, total dominating) set of $D$ is the domination (respectively, total domination) number of $D$, denoted $γ(D)$ (respectively,$γ_t(D)$). The maximum number of pairwise disjoint closed (respectively,open) in-neighborhoods of $D$ is denoted by $ρ(D)$ (respectively,$ρ^{\rm o}(D)$). We prove that in digraphs whose underlying graphs hav…
Dividing sums of cycles in the semiring of functional digraphs
本文研究了在功能有向图的半环中，当B为循环和时，如何以多项式时间解决A整除B的问题，并改进了计算非同构解X的算法。
When all directed cycles have length three
本文针对所有有向环长度均为三的有向图，提出一种构造方法，完整刻画了这类有向图的结构。
An antidirected trail in a digraph is a trail (a walk with no arc repeated) in which the arcs alternate between forward and backward arcs. An antidirected path is an antidirected trail where no vertex is repeated. We show that it is NP-complete to decide whether two vertices $x,y$ in a digraph are connected by an antidirected path, while one can decide in linear time whether they are connected by an antidirected trail. A digraph $D$ is antistrong if it contains an antidirected $(x,y)$-trail starting and ending with a forward arc for every choice of $x,y\in V(D)$. We show that antistrong connec…
Well-Quasi-Ordering Eulerian Digraphs Embeddable in Surfaces by Strong Immersion
本文证明可嵌入任意曲面的最大度至多 4 的欧拉有向图在强浸入下良拟序，并给出圆图、算法与 Erdős-Pósa 性质等应用。
The problem of when a given digraph contains a subdivision of a fixed digraph $F$ is considered. Bang-Jensen et al. laid out foundations for approaching this problem from the algorithmic point of view. In this paper we give further support to several open conjectures and speculations about algorithmic complexity of finding $F$-subdivisions. In particular, up to 5 exceptions, we completely classify for which 4-vertex digraphs $F$, the $F$-subdivision problem is polynomial-time solvable and for which it is NP-complete. While all NP-hardness proofs are made by reduction from some version of the 2…
Reflexive Digraph Reconfiguration by Orientation Strings
针对环到自反有向图环的同态重构，论文用定向串关系刻画重构图连通分量，将复杂度由贪心算法的 Ω(m²) 改进为线性时间、对数空间。
We study convergence to stationarity for random walks on dynamic random digraphs with given degree sequences. The digraphs undergo full regeneration at independent geometrically distributed random time intervals with parameter $α$. Relaxation to stationarity is the result of a competition between regeneration and mixing on the static digraph. When the number of vertices $n$ tends to infinity and the parameter $α$ tends to zero, we find three scenarios according to whether $α\log n$ converges to zero, infinity or to some finite positive value: when the limit is zero, relaxation to stationarity …
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…
Let $X$ be a lazy random walk on a graph $G$. If $G$ is undirected, then the mixing time is upper bounded by the maximum hitting time of the graph. This fails for directed chains, as the biased random walk on the cycle $\mathbb{Z}_n$ shows. However, we establish that for Eulerian digraphs, the mixing time is $O(mn)$, where $m$ is the number of edges and $n$ is the number of vertices. In the reversible case, the mixing time is robust to the change of the laziness parameter. Surprisingly, in the directed setting the mixing time can be sensitive to such changes. We also study exploration and cove…
We prove that for every digraph $F$ and every assignment of pairs of integers $(r_e,q_e)_{e \in A(F)}$ to its arcs there exists an integer $N$ such that every digraph $D$ with dichromatic number at least $N$ contains a subdivision of $F$ in which $e$ is subdivided into a directed path of length congruent to $r_e$ modulo $q_e$, for every $e \in A(F)$. This generalizes to the directed setting the analogous result by Thomassen for undirected graphs, and at the same time yields a novel short proof of his result.
We consider the generalised PageRank walk on a digraph $G$, with refresh probability $α$ and resampling distribution $λ$. We analyse convergence to stationarity when $G$ is a large sparse random digraph with given degree sequences, in the limit of vanishing $α$. We identify three scenarios: when $α$ is much smaller than the inverse of the mixing time of $G$ the relaxation to equilibrium is dominated by the simple random walk and displays a cutoff behaviour; when $α$ is much larger than the inverse of the mixing time of $G$ on the contrary one has pure exponential decay with rate $α$; when $α$ …
We prove the following conjecture of S. Thomassé: for every (potentially infinite) digraph $ D $ it is possible to iteratively reverse directed cycles in such a way that the dichromatic number of the final reorientation $ D^{*} $ of $ D $ is at most two and each edge is flipped only finitely many times. In addition, we guarantee that in every strong component of $ D^{*} $ all the local edge-connectivities are finite and any edge is reversed at most twice.
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。
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 random k-out mapping (digraph) on [n] is generated by choosing k random images of each vertex one at a time, subject to a "preferential attachment" rule: the current vertex selects an image i with probability proportional to a given parameter α= α(n) plus the number of times i has already been selected. Intuitively, the larger αgets, the closer the resulting k-out mapping is to the uniformly random k-out mapping. We prove that α= Θ(n^{1/2}) is the threshold for αgrowing "fast enough" to make the random digraph approach the uniformly random digraph in terms of the total variation distance. We…
In the directed detour problem one is given a digraph $G$ and a pair of vertices $s$ and~$t$, and the task is to decide whether there is a directed simple path from $s$ to $t$ in $G$ whose length is larger than $\mathsf{dist}_{G}(s,t)$. The more general parameterized variant, directed long detour, asks for a simple $s$-to-$t$ path of length at least $\mathsf{dist}_{G}(s,t)+k$, for a given parameter $k$. Surprisingly, it is still unknown whether directed detour is polynomial-time solvable on general digraphs. However, for planar digraphs, Wu and Wang~[Networks, '15] proposed an $\mathcal{O}(n^3…
We study properties of a simple random walk on the random digraph D_{n,p} when np={d\log n},\; d>1. We prove that whp the stationary probability pi_v of a vertex v is asymptotic to deg^-(v)/m where deg^-(v) is the in-degree of v and m=n(n-1)p is the expected number of edges of D_{n,p}. If d=d(n) tends to infinity with n, the stationary distribution is asymptotically uniform whp. Using this result we prove that, for d>1, whp the cover time of D_{n,p} is asymptotic to d\log (d/(d-1))n\log n. If d=d(n) tends to infinity with n, then the cover time is asymptotic to n\log n.
Strong chordality in tournaments and multipartite tournaments with possible loops
本文研究可能带环的多部锦标赛的强弦性，给出多项式时间识别算法与禁用导出子图刻画，并证明强弦有向图中全控制集最小规模等于不相交入邻域最大数目且可在线性时间内计算。
We introduce the notion of z-topological orderings for digraphs. We prove that given a digraph G on n vertices admitting a z-topological order- ing, together with such an ordering, one may count the number of subgraphs of G that at the same time satisfy a monadic second order formula ϕ and are the union of k directed paths, in time f (ϕ, k, z) * n^O(k*z) . Our result implies the polynomial time solvability of many natural counting problems on digraphs admitting z-topological orderings for constant values of z and k. Concerning the relationship between z-topological orderability and other digra…
Given a digraph $G$, a set $X\subseteq V(G)$ is said to be absorbing set (resp. dominating set) if every vertex in the graph is either in $X$ or is an in-neighbour (resp. out-neighbour) of a vertex in $X$. A set $S\subseteq V(G)$ is said to be an independent set if no two vertices in $S$ are adjacent in $G$. A kernel (resp. solution) of $G$ is an independent and absorbing (resp. dominating) set in $G$. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph $G$ is an interval digraph if a pair of intervals $(S_u,T_u)$ can be assigned to e…
We give an "excluded minor" and a "structural" characterization of digraphs D that have the property that for every subdigraph H of D, the maximum number of disjoint circuits in H is equal to the minimum cardinality of a subset T of V(H) such that H\T is acyclic.
We consider the following question. We are given a dense digraph $D$ with minimum in- and out-degree at least $αn$, where $α>1/2$ is a constant. The edges of $D$ are given edge costs $C(e),e\in E(D)$, where $C(e)$ is an independent copy of the uniform $[0,1]$ random variable $U$. Let $C(i,j),i,j\in[n]$ be the associated $n\times n$ cost matrix where $C(i,j)=\infty$ if $(i,j)\notin E(D)$. We show that w.h.p. the patching algorithm of Karp finds a tour for the asymmetric traveling salesperson problem that is asymptotically equal to that of the associated assignment problem. Karp's algorithm runs…
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