Galen E. Turner
Dirac proved that any graph with minimum vertex degree $δ$ contains either a cycle of length at least $2δ$ or a Hamilton cycle. Motivated by this result, we characterize those graphs having no cycle longer than $2δ$.
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Lovász (1965) characterized graphs without two vertex-disjoint cycles, which implies that such graphs have at most three vertices hitting all cycles. In this paper, we ask whether such a small hitting set exists for $S$-cycles, when a graph has no two vertex-disjoint $S$-cycles. For a graph $G$ and a vertex set $S$ of $G$, an $S$-cycle is a cycle containing a vertex of $S$. We provide an example $G$ on $21$ vertices where $G$ has no two vertex-disjoint $S$-cycles, but three vertices are not sufficient to hit all $S$-cycles. On the other hand, we show that four vertices are enough to hit all $S…
Lovász has completely characterised the structure of graphs with no two vertex-disjoint cycles, while Slilaty has given a structural characterisation of graphs with no two vertex-disjoint odd cycles; his result is in fact more general, describing signed graphs with no two vertex-disjoint negative cycles. A biased graph is a graph with a distinguished set of cycles (called balanced) with the property that any theta subgraph does not contain exactly two balanced cycles. In this paper we characterise the structure of biased graphs with no two vertex-disjoint unbalanced cycles, answering a questio…
We investigate minimum vertex degree conditions for $3$-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections consist of precisely one vertex. We prove that every $3$-uniform $n$-vertex ($n$ even) hypergraph $\mathcal{H}$ with minimum vertex degree $δ_1(\mathcal{H})\geq \left(\frac7{16}+o(1)\right)\binom{n}{2}$ contains a loose Hamilton cycle. This bound is asymptotically best possible.
In 1979 Babai found a clever argument to prove that every connected vertex transitive graph on $n \ge 3$ vertices contains a cycle of length at least $\sqrt{3n}$. Here we modify his approach to show that such graphs must contain a cycle of length at least $(1 - o(1))n^{3/5}$.
We prove that if $G$ is a 2-connected graph with $δ(G) \geqslant \frac{v(G) + 2}{3}$ then $G$ has a cycle $W$ such that $V(G - W)$ is independent. This result is best possible in the sense that it becomes false if $\frac{v(G) + 2}{3}$ is replaced by any smaller number.
In this paper, we consider a general degree sum condition sufficient to imply the existence of $k$ vertex-disjoint chorded cycles in a graph $G$. Let $σ_t(G)$ be the minimum degree sum of $t$ independent vertices of $G$. We prove that if $G$ is a graph of sufficiently large order and $σ_t(G)\geq 3kt-t+1$ with $k\geq 1$, then $G$ contains $k$ vertex-disjoint chorded cycles. We also show that the degree sum condition on $σ_t(G)$ is sharp. To do this, we also investigate graphs without chorded cycles.
The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. Bessy, Sereni and Lichiardopol proved that a regular tournament $T$ of minimum degree $2r-1$ contains at least $r$ vertex-disjoint directed cycles, which shows that the above conjecture is true for tournaments. After that, Lichiardopol improved this result by showing that a $2r-1$-regular tournament contains at least $\frac{7}{6}r-\frac{7}{3}$ vertex-disjoint directed cycles. In this paper, we will extend the result t…
We prove that any $3$-uniform hypergraph whose minimum vertex degree is at least $\left(\frac{5}{9} + o(1) \right)\binom{n}{2}$ admits an almost-spanning tight cycle, that is, a tight cycle leaving $o(n)$ vertices uncovered. The bound on the vertex degree is asymptotically best possible. Our proof uses the hypergraph regularity method, and in particular a recent version of the hypergraph regularity lemma proved by Allen, Böttcher, Cooley and Mycroft.
A step forward is made in a long standing Lovász's problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes, other than the Petersen graph, contains a Hamilton cycle. Essential tools used in the proof range from classical results on existence of Hamilton cycles, such as Chvátal's theorem and Jackson's theorem, to certain results on polynomial representations of quadratic residues at primitive roots in finite fields.
A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length $3$ with the edges assigned a same color. It is known that every edge-colored complete graph without containing monochromatic triangles always contains a properly colored Hamilton path. In this paper, we investigate the existence of properly colored cycles in edge-colored complete graphs when monochromatic triangles are forbidden. We obtain a vertex-pancyclic analogous result combined with a characterization of all the excepti…
In 1963, Corrádi and Hajnal settled a conjecture of Erdős by proving that, for all $k \geq 1$, any graph $G$ with $|G| \geq 3k$ and minimum degree at least $2k$ contains $k$ vertex-disjoint cycles. In 2008, Finkel proved that for all $k \geq 1$, any graph $G$ with $|G| \geq 4k$ and minimum degree at least $3k$ contains $k$ vertex-disjoint chorded cycles. Finkel's result was strengthened by Chiba, Fujita, Gao, and Li in 2010, who showed, among other results, that for all $k \geq 1$, any graph $G$ with $|G| \geq 4k$ and minimum Ore-degree at least $6k-1$ contains $k$ vertex-disjoint cycles. We r…
We show that every 3-uniform hypergraph with $n$ vertices and minimum vertex degree at least $(5/9+o(1))\binom{n}2$ contains a tight Hamiltonian cycle. Known lower bound constructions show that this degree condition is asymptotically optimal.
Vertex spans of multilayered cycles
本文定义了多层图及其子类多层圈，并计算了其顶点跨度（强、直接或笛卡尔），发现跨度值仅取决于圈的长度而与层数无关。
The vertex arboricity $a(G)$ of a graph $G$ is the minimum $k$ such that $V(G)$ can be partitioned into $k$ sets where each set induces a forest. For a planar graph $G$, it is known that $a(G)\leq 3$. In two recent papers, it was proved that planar graphs without $k$-cycles for some $k\in\{3, 4, 5, 6, 7\}$ have vertex arboricity at most 2. For a toroidal graph $G$, it is known that $a(G)\leq 4$. Let us consider the following question: do toroidal graphs without $k$-cycles have vertex arboricity at most 2? It was known that the question is true for k=3, and recently, Zhang proved the question i…
The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. In 2014, Bang-Jensen, Bessy and Thomass\' e proved the conjecture for tournaments. In 2010, Lichiardopol conjectured that a tournament $T$ with minimum out-degree at least $(q-1)r-1$ contains at least $r$ vertex-disjoint $q$-cycles, where integer $q\geq3$ and $r\geq1$. In this paper, we address Lichiardopol's conjecture affirmatively. In particular, the case $q=3$ implies Bermond-Thomassen conjecture for tournaments.
Hamilton cycles in vertex-transitive graphs of order $6p$
本文研究阶为 $6p$（$p$ 为素数）的连通顶点传递图，证明除 Petersen 图的三角形替换图外，此类图均包含哈密顿圈，从而加强了已有哈密顿路径结果。
It is conjectured that every edge-colored complete graph $G$ on $n$ vertices satisfying $Δ^{mon}(G)\leq n-3k+1$ contains $k$ vertex-disjoint properly edge-colored cycles. We confirm this conjecture for $k=2$, prove several additional weaker results for general $k$, and we establish structural properties of possible minimum counterexamples to the conjecture. We also reveal a close relationship between properly edge-colored cycles in edge-colored complete graphs and directed cycles in multi-partite tournaments. Using this relationship and our results on edge-colored complete graphs, we obtain se…
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。
We show that for sufficiently large $n$, every 3-uniform hypergraph on $n$ vertices with minimum vertex degree at least $\binom{n-1}2 - \binom{\lfloor\frac34 n\rfloor}2 + c$, where $c=2$ if $n\in 4\mathbb{N}$ and $c=1$ if $n\in 2\mathbb{N}\setminus 4\mathbb{N}$, contains a loose Hamilton cycle. This degree condition is best possible and improves on the work of Buß, Hàn and Schacht who proved the corresponding asymptotical result.
In \cite{listnoC3adjC4}), Borodin and Ivanova proved that every planar graph without $4$-cycles adjacent to $3$-cycle is list vertex-$2$-aborable. In fact, they proved a more general result. Inspired by these results and DP-coloring which becomes a widely studied topic, we introduce a generalization on variable degeneracy including list vertex arboricity. We use this notion to extend a general result by Borodin and Ivanova. Not only that this theorem implies results about planar graphs without $4$-cycles adjacent to $3$-cycle by Borodin and Ivanova, it implies many other results including a re…
On Hamilton cycles in connected vertex-transitive graphs of order $2pq$
本文针对阶为$2pq$且满足特定自同构子群条件的连通点传递图，证明其均含哈密顿圈，为解决代数图论中相关公开问题提供了重要进展。
Answering a question of Häggkvist and Scott, Verstraëte proved that every sufficiently large graph with average degree at least $k^2+19k+10$ contains $k$ vertex-disjoint cycles of consecutive even lengths. He further conjectured that the same holds for every graph $G$ with average degree at least $k^2+3k+2$. In this paper we prove this conjecture for $k\geq 19$ when $G$ is sufficiently large. We also show that for any $ε>0$ and large $k\geq k_ε$, average degree at least $k^2+3k-2+ε$ suffices, which is asymptotically tight for infinitely many graphs.
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\(\Omega(\sqrt n)\) 。
Intersections of longest cycles in vertex-transitive and highly connected graphs
针对顶点传递图和高度连通图中最长圈交集问题，提出纯组合方法强化现有界，证明k-连通图中最长圈交集顶点数达Ω(k^(2/3))，解决Babai问题并引入新构造方法。
Hamilton Cycles In Vertex-Transitive Graphs of Order 10p
本文证明了每个10p阶（p为素数）连通点传递图都包含哈密顿圈，唯一例外是Petersen图的截断，从而推进了该类图的哈密顿性问题研究。
A fan $F_k$ is a graph that consists of an induced path on $k$ vertices and an additional vertex that is adjacent to all vertices of the path. We prove that for all positive integers $q$ and $k$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a vertex-minor isomorphic to $F_k$. We also prove that for all positive integers $q$ and $k\ge 3$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a pivot-minor isomorphic to a cycle of length $k$.
Let $n\geq 6,k\geq 0$ be two integers. Let $H$ be a graph of order $n$ with $k$ components, each of which is an even cycle of length at least $6$ and $G$ be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|\geq n/2$. In this paper, we show that if the minimum degree of $G$ is at least $n/2-k+1$, then $G$ contains a subdivision of $H$. This generalized an older result of Wang.
Two-disjoint-cycle-cover vertex pancyclicity of split-star networks
本文研究分裂星网络S_n^2的两不交圈覆盖顶点泛圈性，证明对于n≥4，S_n^2是2-DCC顶点[3,n!/2]-泛圈的。
It is known that every strong tournament has directed cycles of any length, and thereby strong subtournaments of any size. In this note, we prove that they also can share a common vertex which is a king of all of them. This common vertex can be any king in the whole tournament. Further, the Hamiltonian cycles in them can be recursively constructed by inserting an additional vertex to one directed edge.
Ryser proved that any two tournaments with the same score sequence are $C_3$-equivalent while Beineke and Moon proved the $C_4$-equivalence for any two bipartite tournaments with the same score lists. In this paper, we extend these results to orientations of $G$ vertex-multiplications. We focus on two main areas, namely orientations with the same score list and with score-list parity. Our main tools are extensions of the refinement technique, directed difference graph and a reduction lemma.
On vertex-transitive graphs with a unique hamiltonian cycle
本文研究无限顶点传递图中的唯一哈密顿性，分类了有限端点的唯一哈密顿顶点传递图，并构造出每个非阿贝尔自由群上具有唯一哈密顿圆且外可平面的凯莱图。
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 …
We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. A sufficient condition for vertex adjacency in the 1-skeleton of the traveling salesperson polytope can be formulated as the Hamiltonian decomposition problem in a 4-regular multigraph. We introduce a heuristic general variable neighborhood search algorithm for this problem based on finding a vertex-disjoint cycle cover of the multigraph through reduction to perfect matching and several cycle merging operations. The algorithm has a one-sided error: the answer "not adjacent" is…
A set $S$ of vertices in a graph $G = (V, E)$ is called {\em cycle independent} if the induced subgraph $\langle S\rangle$ is acyclic, and called {\em odd-cycle indepdendet} if $\langle S\rangle$ is bipartite. A set $S$ is {\em cycle dominating} (resp. {\em odd-cycle dominating}) if for every vertex $u \in V \setminus S$ there exists a vertex $v \in S$ such that $u$ and $v$ are contained in a (resp. odd cycle) cycle in $\langle S \setminus \{u\}\rangle$. A set $S$ is {\em cycle irredundant} (resp. odd-cycle irredundant) if for every vertex $v \in S$ there exists a vertex $u \in V \setminus S$ …
Spanning cycles in random directed graphs
研究随机有向图中生成圈的出现时间，证明任意有向圈几乎必然早于有向哈密顿圈出现，并确定其尖锐阈值，证实了 Ferber-Long 猜想。
Edge-disjoint cycles with the same vertex set
本文研究Erdős关于同顶点集边不交圈的极值问题，证明边数达n·polylog(n)的图必含k个此类圈，并给出匹配的多对数下界及新正则化工具。
The Turán number of a $k$-uniform hypergraph $H$, denoted by $e{x_k}\left({n;H} \right)$, is the maximum number of edges in any $k$-uniform hypergraph $F$ on $n$ vertices which does not contain $H$ as a subgraph. Let $\mathcal{C}_{\ell}^{\left(k \right)}$ denote the family of all $k$-uniform minimal cycles of length $\ell$, $\mathcal{S}(\ell_1,\ldots,\ell_r)$ denote the family of hypergraphs consisting of unions of $r$ vertex disjoint minimal cycles of length $\ell_1,\ldots,\ell_r$, respectively, and $\mathbb{C}_{\ell}^{\left(k \right)}$ denote a $k$-uniform linear cycle of length $\ell$. We d…
In order to complete (and generalize) results of Gardiner and Praeger on 4-valent symmetric graphs (European J. Combin, 15 (1994)) we apply the method of lifting automorphisms in the context of elementary-abelian covering projections. In particular, the vertex- and edge-transitive graphs whose quotient by a normal $p$-elementary abelian group of automorphisms, for $p$ an odd prime, is a cycle, are described in terms of cyclic and negacyclic codes. Specifically, the symmetry properties of such graphs are derived from certain properties of the generating polynomials of cyclic and negacyclic code…
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 …
In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently, a disjoint union of paths). Note that we can partition the vertex set of a forest into two independent sets. However, a pair of independent sets combined may not induce a forest. Thus our result extends the result of Wang and Xu (2013) stating that the vertex set of every planar graph without $4$-cycles and $6$-cycles can be partitioned into three sets, wher…
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 $k\le 2$.
A tight $k$-uniform $\ell$-cycle, denoted by $TC_\ell^k$, is a $k$-uniform hypergraph whose vertex set is $v_0, \cdots, v_{\ell-1}$, and the edges are all the $k$-tuples $\{v_i, v_{i+1}, \cdots, v_{i+k-1}\}$, with subscripts modulo $\ell$. Motivated by a classic result in graph theory that every $n$-vertex cycle-free graph has at most $n-1$ edges, Sós and, independently, Verstraëte asked whether for every integer $k$, a $k$-uniform $n$-vertex hypergraph without any tight $k$-uniform cycles has at most $\binom{n-1}{k-1}$ edges. In this paper, we answer this question in negative.
An edge colouring of a graph is said to be an $r$-local colouring if the edges incident to any vertex are coloured with at most $r$ colours. Generalising a result of Bessy and Thomassé, we prove that the vertex set of any $2$-locally coloured complete graph may be partitioned into two disjoint monochromatic cycles of different colours. Moreover, for any natural number $r$, we show that the vertex set of any $r$-locally coloured complete graph may be partitioned into $O(r^2 \log r)$ disjoint monochromatic cycles. This generalises a result of Erdős, Gyárfás and Pyber.
In 1975, Pippenger and Golumbic conjectured that every n-vertex graph has at most $n^k/(k^k - k)$ induced cycles of length k for k at least 5. We prove that every n-vertex graph has at most $2 n^k/k^k$ induced cycles of length k.
It is shown that every connected vertex-transitive graph of order $6p$, where $p$ is a prime, contains a Hamilton path. Moreover, it is shown that, except for the truncation of the Petersen graph, every connected vertex-transitive graph of order $6p$ which is not genuinely imprimitive contains a Hamilton cycle.
Counting triangles in graphs without vertex disjoint odd cycles
本文确定了在禁止包含 $\ell+1$ 个顶点不相交奇圈的图中三角形副本的最大数量及其极图，推广了已有结论。
After long term efforts, it was recently proved in \cite{DKM2} that except for the Peterson graph, every connected vertex-transitive graph of order $rs$ has a Hamilton cycle, where $r$ and $s$ are primes. A natural topic is to solve the hamiltonian problem for connected vertex-transitive graphs of $2rs$. This topic is quite trivial, as the problem is still unsolved even for that of $r=3$. In this paper, it is shown that except for the Coxeter graph, every connected vertex-transitive graph of order $2rs$ contains a Hamilton cycle, provided the automorphism group acts primitively on vertices.
We prove that for every graph, any vertex subset $S$, and given integers $k,\ell$: there are $k$ disjoint cycles of length at least $\ell$ that each contain at least one vertex from $S$, or a vertex set of size $O(\ell \cdot k \log k)$ that meets all such cycles. This generalises previous results of Fiorini and Hendrickx and of Pontecorvi and Wollan. In addition, we describe an algorithm for our main result that runs in $O(k \log k \cdot s^2 \cdot (f(\ell) \cdot n+m))$ time, where $s$ denotes the cardinality of $S$.
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。
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…
In a graph G; a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S: The cardinality of a smallest vertex (resp. edge) metric generator is the vertex (resp. edge) metric dimension of G: In [19] we determined the vertex (resp. edge) metric dimension of unicyclic graphs and that it takes its value from two consecutive integers. Therein, several cycle configurations were introduced and the vertex (resp. edge) metric dimension takes the greater of the two consecutive values only if any of …
We show that under certain conditions the square of the graph obtained by identifying a vertex in two graphs with hamiltonian square is also hamiltonian. Using this result, we prove necessary and sufficient conditions for hamiltonicity of the square of a connected graph such that every vertex of degree at least three in a block graph corresponds to a cut vertex and any two these vertices are at distance at least four.
Consider a simple graph in which a random walk begins at a given vertex. It moves at each step with equal probability to any neighbor of its current vertex, and ends when it has visited every vertex. We call such a random walk a random cover tour. It is well known that cycles and complete graphs have the property that a random cover tour starting at any vertex is equally likely to end at any other vertex. Ronald Graham asked whether there are any other graphs with this property. In 1993, Lászlo Lovász and Peter Winkler showed that cycles and complete graphs are the only undirected graphs with …
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 …
We determine the maximum number of edges in a $K_4$-minor-free $n$-vertex graph of girth $g$, when $g = 5$ or $g$ is even. We argue that there are many different $n$-vertex extremal graphs, if $n$ is even and $g$ is odd.
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。
Let G be a graph whose edges are labeled by positive integers. Label each vertex with an integer and suppose if two vertices are joined by an edge, the vertex labels are congruent to each other modulo the edge label. The set of vertex labels satisfying this condition is called a generalized spline. Gilbert, Polster, and Tymoczko recently defined generalized splines based on work on polynomial splines by Billera, Rose, Haas, Goresky-Kottwitz-Machperson, and many others. We focus on generalized splines on $n$-cycles. We construct a particularly nice basis for the module of splines on $n$-cycles.…
An Italian dominating function on a (di)graph $G$ with vertex set $V(G)$ is a function $f: V(G) \to \{0, 1, 2\}$ such that every vertex $v \in V(G)$ such that $f(v) = 0$ has an (in)neighbour assigned 2 or two (in)neighbours assigned 1. We complete the investigation of the Italian domination numbers of Cartesian products of directed cycles.
The Wiener index of vertex colorings
本文研究路径和圈上顶点染色的Wiener指标，刻画了使该指标最大的染色，并建立了优超序与Wiener指标的联系。
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。
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