Peter Keevash, Daniela Kühn, Deryk Osthus
We show that every sufficiently large oriented graph with minimum in- and outdegree at least (3n-4)/8 contains a Hamilton cycle. This is best possible and solves a problem of Thomassen from 1979.
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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。
We prove that a tournament and its complement contain the same number of oriented Hamiltonian paths (resp. cycles) of any given type, as a generalization of Rosenfeld's result proved for antidirected paths.
An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle $C$, there are digraphs containing no subdivision of $C$ (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any $C$ a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of $C$. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out-degree $2$ and two vertices have in-degree $2$).
We use a randomised embedding method to prove that for all α>0 any sufficiently large oriented graph G with minimum in-degree and out-degree δ^+(G),δ^-(G)\geq (3/8+α)|G| contains every possible orientation of a Hamilton cycle. This confirms a conjecture of Häggkvist and Thomason.
We present statistics on the decompositions (with respect to a distinguished symmetric 2t-cycle) of vertices of the hypercube graph, whose negative parts are covered by two subsets of the ground set {1,...,t} of the corresponding oriented matroid.
An oriented 3-graph consists of a family of triples (3-sets), each of which is given one of its two possible cyclic orientations. A cycle in an oriented 3-graph is a positive sum of some of the triples that gives weight zero to each 2-set. Our aim in this paper is to consider the following question: how large can the girth of an oriented 3-graph (on $n$ vertices) be? We show that there exist oriented 3-graphs whose shortest cycle has length $\frac{n^2}{2}(1+o(1))$: this is asymptotically best possible. We also show that there exist 3-tournaments whose shortest cycle has length $\frac{n^2}{3}(1…
Rosenfeld in 1974 conjectured that there is an integer N > 8 such that every tournament of order n > N contains every non-directed cycle of order n. We prove that, with exactly 35 exceptions, every tournament of order n > 2 contains each non-directed cycle of order m, 2 < m < n+1.
We show that for each \ell\geq 4 every sufficiently large oriented graph G with δ^+(G), δ^-(G) \geq \lfloor |G|/3 \rfloor +1 contains an \ell-cycle. This is best possible for all those \ell\geq 4 which are not divisible by 3. Surprisingly, for some other values of \ell, an \ell-cycle is forced by a much weaker minimum degree condition. We propose and discuss a conjecture regarding the precise minimum degree which forces an \ell-cycle (with \ell \geq 4 divisible by 3) in an oriented graph. We also give an application of our results to pancyclicity and consider \ell-cycles in general digraphs.
By using a $q$-analogue of the "magic" matrix introduced by H.Huang in his elegant solution of the sensitivity conjecture, we give a direct generalization of his result, replacing a hypercube graph by a Cartesian power of a directed $l$-cycle.
We present statistics on the decompositions (with respect to a distinguished symmetric 2t-cycle) of vertices of the hypercube graph, whose negative parts are regarded as disjoint unions of two subsets of the ground set {1,...,t} of the corresponding oriented matroid.
Oriented Discrepancy of The Square of Hamilton Cycles
本文研究了在给定最小度数条件下，Hamilton回路的平方是否存在具有大正向与反向边不平衡的子图，证明了当n足够大时，每个具有最小度数δ(G)≥2n/3的定向图G都包含一个Hamilton回路的平方，其最大σ值超过依赖于δ(G)和n的函数。
We propose the following conjecture extending Dirac's theorem: if $G$ is a graph with $n\ge 3$ vertices and minimum degree $δ(G)\ge n/2$, then in every orientation of $G$ there is a Hamilton cycle with at least $δ(G)$ edges oriented in the same direction. We prove an approximate version of this conjecture, showing that minimum degree $n/2 + O(k)$ guarantees a Hamilton cycle with at least $(n+k)/2$ edges oriented in the same direction. We also study the analogous problem for random graphs, showing that if the edge probability $p = p(n)$ is above the Hamiltonicity threshold, then, with high prob…
We present explicit descriptions of the decompositions of vertices of a hypercube graph with respect to its distinguished symmetric cycle.
We show that for each α>0 every sufficiently large oriented graph G with δ^+(G),δ^-(G)\ge 3|G|/8+ α|G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. In fact, we prove the stronger result that G is still Hamiltonian if δ(G)+δ^+(G)+δ^-(G)\geq 3|G|/2 + α|G|. Up to the term α|G| this confirms a conjecture of Häggkvist. We also prove an Ore-type theorem for oriented graphs.
Motivated by his work on the classification of countable homogeneous oriented graphs, Cherlin asked about the typical structure of oriented graphs (i) without a transitive triangle, or (ii) without an oriented triangle. We give an answer to these questions (which is not quite the predicted one). Our approach is based on the recent `hypergraph containers' method, developed independently by Saxton and Thomason as well as by Balogh, Morris and Samotij. Moreover, our results generalise to forbidden transitive tournaments and forbidden oriented cycles of any order, and also apply to digraphs. Along…
Pattern Recognition on Oriented Matroids: Symmetric Cycles in the Hypercube Graphs. VI
本文探讨二元数组精确模型表示的线性代数、组合与应用层面，并以字符串的两幅线性代数画像加以说明。
Kelly, Kuehn and Osthus conjectured that for any l>3 and the smallest number k>2 that does not divide l, any large enough oriented graph G with minimum indegree and minimum outdegree at least \lfloor |V(G)|/k\rfloor +1 contains a directed cycle of length l. We prove this conjecture asymptotically for the case when l is large enough compared to k and k>6. The case when k<7 was already settled asymptotically by Kelly, Kuehn and Osthus.
We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several extremal results. In particular, we show that every nearly $cn$-regular oriented graph on $n$ vertices with $c>3/8$ contains $(cn/e)^n(1+o(1))^n$ directed Hamilton cycles. This is an extension of a result of Cuckler, who settled an old conjecture of Thomassen about the number of Hamilton cycles in regular tournaments. We also prove that every graph $G$ on $n$ vertices of minimum degree at least $(1/2+\varepsilon)n$ con…
We refine two results of Jiang, Shao and Vesel on the $L(2,1)$-labeling number $λ$ of the Cartesian and the strong product of two oriented cycles. For the Cartesian product, we compute the exact value of $λ(\overrightarrow{C_m} \square \overrightarrow{C_n})$ for $m$, $n \geq 40$; in the case of strong product, we either compute the exact value or establish a gap of size one for $λ(\overrightarrow{C_m} \boxtimes \overrightarrow{C_n})$ for $m$, $n \geq 48$.
If V is the vertex sequence of a symmetric 2t-cycle in the hypercube graph with the vertices {1,-1}^t, then for any vertex T of the graph there exists a unique inclusion-minimal subset of V such that T is the sum of its elements. We present a simple combinatorial statistic on decompositions of vertices of the hypercube graphs with respect to symmetric cycles and describe their basic metric properties.
Ramsey--Turán Factors of Non-directed Oriented Cycles in Oriented Graphs
该研究针对有向图，证明满足最小半度阈值及独立数条件的足够大的有向图包含指定长度的反有向环因子，其最小半度阈值渐近紧，采用了Ramsey-Turán型格吸收引理完成证明。
A graph $G$ has a $k$-page book embedding if $G$ can be embedded into a $k$-page book. The minimum $k$ such that $G$ has a $k$-page book embedding is the book thickness of $G$, denoted $bt(G)$. Most of the work on this subject has been done for unoriented graphs and oriented acyclic graphs (no directed cycles). In this work we discuss oriented graphs $\overrightarrow{D}$ containing directed cycles by using oriented book embeddings and oriented book thickness, $obt(\overrightarrow{D})$. To characterize $\overrightarrow{D}$ such that $obt(\overrightarrow{D}) = k$, we define the class $\mathcal{M…
Subdivisions of Oriented Cycles in Digraphs with Hamiltonian directed path
针对有向图含指定有向圈细分的色数猜想，本文改进了哈密顿有向图情形的色数上界至$2n$，并将结果拓展到含哈密顿有向路径的有向图，深化了对猜想条件的认识。
Forward Arc Maximization for Hamilton Oriented Cycles and Paths in Generalizations of Tournaments
本文研究了竞赛图推广中有向Hamilton环和路径中前向弧最大数量的问题，提出了多项式时间算法，并揭示了某些推广中的NP难性。
Arbitrary orientations of cycles in oriented graphs
该研究证明了在特定度数条件下，有向图包含任意定向的Hamilton环，并推导了关于任意定向环的pancyclicity结果。
Powers of Hamilton cycles in oriented and directed graphs
本文研究有向图与定向图中 Hamilton 圈幂的度阈值，渐近确定有向图中 Hamilton 圈平方的最小总度阈值，并给出定向图中保证 k 次幂的半度条件。
A Cambrian framework for the oriented cycle
本文通过扩展可排序元素的概念，为具有有限或仿射型Cartan伴矩阵的交换矩阵构造了寒武纪框架，特别解决了非无圈仿射情形（有向n-圈）的框架构造问题。
We consider decompositions of topes of the oriented matroid realizable as the arrangement of coordinate hyperplanes in $\mathbb{R}^{2^t}$, with respect to a distinguished symmetric $2\cdot 2^t$-cycle in its hypercube graph of topes $\boldsymbol{H}(2^t,2)$. We seek interpretations of such decompositions in the context of subset families on the ground set $E_t:=\{1,\ldots,t\}$ and of the families of their blocking sets, in the context of clutters on $E_t$ and of their blockers.
Universality for rainbow oriented cycles in perturbed digraphs
研究随机扰动有向图在均匀n边着色下，高概率同时包含所有长度和定向的彩虹有向圈，通过Montgomery分布吸收法证明。
Hamilton Cycles in Dense Regular Digraphs and Oriented Graphs
本文证明了度数至少为 $(1/4 + \varepsilon)n$ 的稠密正则定向图包含哈密顿圈，确立了 Jackson 1981 年猜想的近似版本，并拓展了正则有向图哈密顿性的相关结果。
Oriented Hamiltonian Paths in Tournaments: Stability under Arc Deletion
本文研究了竞赛中定向哈密顿路径在任意弧删除下的稳定性，证明了除特定例外外，删除任意一个弧后仍能保持所有定向哈密顿路径的存在。
An exact Ore-degree condition for Hamilton cycles in oriented graphs
本文证明充分大的有向图在满足最优 Ore 型度条件 deg⁺(x)+deg⁻(y)≥(3n−3)/4 时必含哈密顿圈，解决了 Kühn 和 Osthus 的公开问题，并推广和改进了已有结果。
Graph Puzzles I.1: Oriented Berge-Fulkerson Conjecture
本文探讨了有向伯格-福克逊猜想的有向版本，证明了伊萨克斯花状图族的该猜想，并讨论了如何通过有向表面构建6-圈双覆盖。
A Hamiltonian decomposition of $G$ is a partition of its edge set into disjoint Hamilton cycles. Manikandan and Paulraja conjectured that if $G$ and $H$ are Hamilton cycle decomposable circulant graphs with at least one of them is nonbipartite, then their tensor product is Hamilton cycle decomposable. In this paper, we have proved that, if $G$ is a Hamilton cycle decomposable circulant graph with certain properties and $H$ is a Hamilton cycle decomposable multigraph, then their tensor product is Hamilton cycle decomposable. In particular, tensor products of certain sparse Hamilton cycle decomp…
Cycle Partitions in Dense Regular Digraphs and Oriented Graphs
本文证明了Jackson在1981年提出的关于大$d$-正则定向图哈密顿性的猜想，并推广证明了满足$d\geqαn$的稠密$d$-正则有向图和定向图可分别被至多$n/(d+1)$和$n/(2d+1)$个顶点不相交的圈覆盖。
Cycle-factors in oriented graphs
本文研究高最小入度和出度有向图中的圈因子，证明接近 $n/2$ 的度条件可保证任意 $k\geq4$ 的 $k$-圈因子及给定长度的互不相交圈集合，改进了已有结果。
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…
An Ore-type Theorem for Oriented Discrepancy of Hamilton Cycles
本文研究有向图中Hamilton环的定向偏差问题，证明当非相邻顶点度数和σ₂(G)≥n时，图中存在一个Hamilton环，其中至少有n/2或σ₂(G)/2-O(n)条边方向一致。
In this paper we give a complete solution to the Hamilton-Waterloo problem for the case of Hamilton cycles and C4k-factors for all positive integers k.
The generalizations of Hamiltonian in oriented graphs
本文推广了Keevash等人关于高半度有向图哈密顿性的结果，证明其包含指定长度的多个不相交圈，并确定了强哈密顿连通等性质所需的半度界限。
In the Oriented-cycle game, introduced by Bollobás and Szabó, two players, called OMaker and OBreaker, alternately direct edges of $K_n$. OMaker directs exactly one edge, whereas OBreaker is allowed to direct between one and $b$ edges. OMaker wins if the final tournament contains a directed cycle, otherwise OBreaker wins. Bollobás and Szabó conjectured that for a bias as large as $n-3$ OMaker has a winning strategy if OBreaker must take exactly $b$ edges in each round. It was shown recently by Ben-Eliezer, Krivelevich and Sudakov, that OMaker has a winning strategy for this game whenever $b\le…
Ore-type condition for antidirected Hamilton cycles in oriented graphs
本文研究了有向图中反向有向哈密尔顿环的 Ore 型条件，证明了在特定度数条件下存在反向有向哈密尔顿环，并展示了该条件的最优性。
We compute the number of equivalence classes of nonperiodic covering cycles of given length in a non oriented connected graph. A covering cycle is a closed path that traverses each edge of the graph at least once. A special case is the number of Euler cycles in the non oriented graph. An identity relating the numbers of covering cycles of any length in a graph to a product of determinants is obtained.
In 1880, P. G. Tait showed that the four colour theorem is equivalent to the assertion that every 3-regular planar graph without cut-edges is 3-edge-colourable, and in 1891, J. Petersen proved that every 3-regular graph with at most two cut-edges has a 1-factor. In this paper, we introduce the notion of collapsing all edges of a 1-factor of a 3-regular planar graph, thereby obtaining what we call a vertex-oriented 4-regular planar graph. We also introduce the notion of o-colouring a vertex-oriented 4-regular planar graph, and we prove that the four colour theorem is equivalent to the assertion…
Let $G^σ$ be an oriented graph and $S(G^σ)$ be its skew-adjacency matrix, where $G$ is called the underlying graph of $G^σ$. The skew-rank of $G^σ$, denoted by $sr(G^σ)$, is the rank of $S(G^σ)$. Denote by $d(G)=|E(G)|-|V(G)|+θ(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $θ(G)$ are the edge number, vertex number and the number of connected components of $G$, respectively. Recently, Wong, Ma and Tian [European J. Combin. 54 (2016) 76--86] proved that $sr(G^σ)\leq r(G)+2d(G)$ for an oriented graph $G^σ$, where $r(G)$ is the rank of the adjacency matrix of $G$, and char…
For a symmetric 2t-cycle in the tope graph of a simple oriented matroid M on the ground set {1,...,t}, where t is even, we describe decompositions of topes and subtopes of M with respect to the subtopes corresponding to the edges of the symmetric cycle.
On oriented cycles in randomly perturbed digraphs
该研究推广了Bohman等人关于随机扰动有向图中哈密顿圈的结果，证明在最小半度条件下，渐近几乎必然同时存在所有长度和所有定向的圈，并可将条件放宽至最小总度情形，方法基于Montgomery吸收方法的变体。
A Hamilton cycle in a directed graph $G$ is a cycle that passes through every vertex of $G$. A Hamiltonian decomposition of $G$ is a partition of its edge set into disjoint Hamilton cycles. In the late $60$s Kelly conjectured that every regular tournament has a Hamilton decomposition. This conjecture was recently settled by Kühn and Osthus, who proved more generally that every $r$-regular $n$-vertex oriented graph $G$ (without antiparallel edges) with $r=cn$ for some fixed $c>3/8$ has a Hamiltonian decomposition, provided $n=n(c)$ is sufficiently large. In this paper we address the natural que…
If the line graph of a graph $G$ decomposes into Hamiltonian cycles, what is $G$? We answer this question for decomposition into two cycles.
New Ore-type conditions for Hamilton cycles and spanning trees with few leaves are presented.
D-Antimagic Labelings of Oriented 2-Regular Graphs
本文研究有向2-正则图的D-反幻标号，分别在不同距离集大小下刻画了有向圈、单向奇圈、Θ-有向圈及相应2-正则图的D-反幻性质。
We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in ${\cal D}(n,p)$ for nearly optimal $p$ (up to a $\log ^cn$ factor). In particular, we show that given $t = (1-o(1))np$ Hamilton cycles $C_1,\ldots ,C_{t}$, each of which is oriented arbitrarily, a digraph $D \sim {\cal D}(n,p)$ w.h.p. contains edge disjoint copies of $C_1,\ldots ,C_t$, provided $p=ω(\log ^3 n/n)$. We also show that given an arbitrarily oriented $n$-vertex cycle $C$, a random digraph $D \sim {\cal D}(n,p)$ w.h.p. contains $(1\pm o(1))n!p^n$ copies of $C$, provided $p \geq \log ^{1 + o(1)}n/n$.
Investigating a problem of B. Mohar, we show that every one-ended Hamiltonian cubic graph with end degree 3 contains a second Hamilton cycle. We also construct two examples showing that this result does not extend to give a third Hamilton cycle, nor that it extends to the two-ended case.
The Prescribed-Vertex Semidegree Threshold for Directed $3q$-Cycles in Oriented Graphs
该研究确定有向图中3q-有向环的指定顶点半度阈值，填补相关定理的缺口，优化了一般指定顶点定理的阶数假设。
A small oriented cycle double cover (SOCDC)} of a bridgeless graph $G$ on $n$ vertices is a collection of at most $n-1$ directed cycles of the symmetric orientation, $G_s$, of $G$ such that each edge of $G_s$ lies in exactly one of the cycles. It is conjectured that every 2-connected graph except two complete graphs $K_4$ and $K_6$ has an $\rm SOCDC$. In this paper, we study graphs with $\rm SOCDC$ and obtain some properties of the minimal counterexample to this conjecture.
4-regular 4-connected Hamiltonian graphs with few Hamiltonian cycles
该研究证明了存在哈密顿环数量有界的无穷多族4-正则4-连通哈密顿图，并提出5-正则5-连通情形下是否存在同类图族的开放问题。
Necessary condition to have Hamiltonian cycle in planar graph is given. Examples of regular planar graphs degree three without Hamiltonian cycle are built.
Akbari, Etesami, Mahini, and Mahmoody conjectured that every proper edge colouring of $K_n$ with $n$ colours contains a Hamilton cycle with $\leq O(\log n)$ colours. They proved that there is always a Hamilton cycle with $\leq 8\sqrt n$ colours. In this note we improve this bound to $O(\log^3 n)$.
Counting traversing Hamiltonian cycles in tiled graphs
该研究把遍历 Hamilton 圈计数推广到广义平铺图，并证明固定瓦片集合下可按图规模线性时间计数，属于固定参数可处理问题。
We prove new lower bounds on the crossing number of a complete graphs assuming that it is drawn in such a way that it contains a Hamiltonian cycle with no crossings.
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