Howard Kleiman
We give polynomial-time algorithms for obtaining hamilton circuits in random graphs, G, and random directed graphs, D. If n is finite, we assume that G or D contains a hamilton circuit. If G is an arbitrary graph containing a hamilton circuit, we conjecture that Algorithm G always obtains a hamilton circuit in polynomial time.
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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.
Eulerian Directed Multigraphs
本文针对有限连通非平凡有向多重图，证明了存在欧拉有向回路与非回路有向路径的充要条件，即顶点对的共回路性与特定顶点出入度关系同时成立。
Decompositions of the wreath product of certain directed graphs into directed hamiltonian cycles
本文针对 Alspach 等人提出的圈积保持哈密顿分解性猜想，证明偶数阶可分解有向图与较长有向圈或完全对称有向图的圈积可哈密顿分解，并给出与短圈圈积不可分解的反例。
In 1960, Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if D is a directed graph on n vertices with minimum out-degree and in-degree at least n/2 (i.e. minimum semi-degree at least n/2), then D contains a directed Hamiltonian cycle. Of course there are other orientations of a cycle in a directed graph and it is not clear that the semi-degree threshold for the directed Hamiltonian cycle is the same as the semi-degree threshold for some other orientation. In 1980, Grant initiated the problem of determining the minimum semi-degree threshold for the anti-directed Hamilto…
Let $\{D_M\}_{M\geq 0}$ be the $n$-vertex random directed graph process, where $D_0$ is the empty directed graph on $n$ vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each $\varepsilon>0$, we show that, almost surely, any directed graph $D_M$ with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most $(1/2-\varepsilon)$ of both the in- and out-edges incident to each vertex are removed. We say such a directed grap…
A classical theorem of Ghouila-Houri from 1960 asserts that every directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$ contains a directed Hamilton cycle. In this paper we extend this theorem to a random directed graph ${\mathcal D}(n,p)$, that is, a directed graph in which every ordered pair $(u,v)$ becomes an arc with probability $p$ independently of all other pairs. Motivated by the study of resilience of properties of random graphs, we prove that if $p \gg \log n/\sqrt{n}$, then a.a.s. every subdigraph of ${\mathcal D}(n,p)$ with minimum out-degree and in-deg…
Let D(n,p) be the random directed graph on n vertices where each of the n(n-1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if $p\ge(\log n+ω(1))/n$ then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically $n!(p(1+o(1)))^{n}$. We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in…
The Directed Hamilton-Waterloo Problem asks for a directed $2$-factorization of the complete symmetric digraph $K_v^*$ where there are two non-isomorphic $2$-factors. In the uniform version of the problem, factors consist of either directed $m$-cycles or $n$-cycles. In this paper, necessary conditions for a solution of this problem are given, and the problem is solved for the factors with $(m, n)\in \{(4,6),(4,8),(4,12),(4,16),(6,12)$,$(8,16),(3,5),(3,15),(5,15)\}$.
We survey some recent results on long-standing conjectures regarding Hamilton cycles in directed graphs, oriented graphs and tournaments. We also combine some of these to prove the following approximate result towards Kelly's conjecture on Hamilton decompositions of regular tournaments: the edges of every regular tournament can be covered by a set of Hamilton cycles which are `almost' edge-disjoint. We also highlight the role that the notion of `robust expansion' plays in several of the proofs. New and old open problems are discussed.
Given a 0-1 square matrix A, when can some of the 1's be changed to -1's in such a way that the permanent of A equals the determinant of the modified matrix? When does a real square matrix have the property that every real matrix with the same sign pattern (that is, the corresponding entries either have the same sign or are both zero) is nonsingular? When is a hypergraph with n vertices and n hyperedges minimally nonbipartite? When does a bipartite graph have a "Pfaffian orientation"? Given a digraph, does it have no directed circuit of even length? Given a digraph, does it have a subdivision …
A Hamilton cycle in a digraph is a cycle that passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this, is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so called Pos?a `rotation-extension' technique for the undirected analogue. Let ${\mathcal D}(n,p)$ denote the random digraph on vertex set $[n]$, obtained by adding each directed edge independently with probability $p$. Here, we prese…
Our studies are related to a special class of FASS-curves, which can be described in a node-rewriting Lindenmayer-system. These ortho-tile (or diagonal) type recursive curves inducing Hamiltonian paths. We define a special directed graph on a rectangular grid, and we enumerate all Hamiltonian paths on this graph. Our formulas are strongly related to both the Fibonacci numbers and the domino tilings of chessboards. The constructability of the regular $17$-gon with straightedge and compass is also related.
Arbitrary orientations of Hamilton cycles in directed graphs of large minimum degree
本文将 Ghouila-Houri 的有向 Hamilton 圈定理渐近推广到任意圈定向，证明最小度至少为 (1+o(1))n 的有向图包含所有可能长度圈的各种定向，仅非强连通情形下有向 Hamilton 圈例外。
Hamiltonian decompositions of the wreath product of hamiltonian decomposable digraphs
本文证实了关于两个哈密顿可分解有向图的圈积仍为哈密顿可分解的猜想的大部分情形，并给出了唯一的可能例外条件。
The Directed Uniform Hamilton-Waterloo Problem Involving Even Cycle Sizes
本文研究完全对称有向图的均匀分解问题，推广无向Hamilton-Waterloo问题，证明除少数例外，可分解为两个非同构均匀因子，涉及有向偶圈。
Consider a directed analogue of the random graph process on $n$ vertices, where the $n(n-1)$ edges are ordered uniformly at random and revealed one at a time. It is known that w.h.p.\@ the first digraph in this process with both in-degree and out-degree $\geq q$ has a $[q]$-edge-coloring with a Hamilton cycle in each color. We show that this coloring can be constructed online, where each edge must be irrevocably colored as soon as it appears. In a similar fashion, for the \emph{undirected} random graph process, we present an online $[n]$-edge-coloring algorithm which yields w.h.p.\@ $q$ disjoi…
In 1972, Woodall raised the following Ore type condition for directed Hamilton cycles in digraphs: Let $D$ be a digraph. If for every vertex pair $u$ and $v$, where there is no arc from $u$ to $v$, we have $d^+u)+d^-(v)\geq |D|$, then $D$ has a directed Hamilton cycle. By a correspondence between bipartite graphs and digraphs, the above result is equivalent to the following result of Las Vergnas: Let $G = (B,W)$ be a balanced bipartite graph. If for any $b \in B$ and $w \in W$, where $b$ and $w$ are nonadjacent, we have $d(w)+d(b) \geq |G|/2 + 1$, then every perfect matching of $G$ is containe…
Consider the random graph process where we start with an empty graph on n vertices, and at time t, are given an edge e_t chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that w.h.p. the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph has a directed Hamilton cycle w.h.p. only at time (1+o(1))n log n. In this paper we further study the directed case, and ask whether it is essential to have twice as many edges compared to the undirected case. Mo…
In this paper, the leading term of the asymptotics of the number of possible final positions of a random walk on a directed Hamiltonian metric graph is found. Consideration of such dynamical systems could be motivated by problems of propagation of narrow wave packets on metric graphs.
Directed Hamiltonicity in Generalized Kneser Graphs
本文证明广义Kneser图$KG(n,k,s)$的规范定向包含有向哈密顿圈，并确定其双色数为$k$，通过将Ledezma和Pastine的类图框架推广到有向情形，利用循环旋转和友类邻接构造了跨越所有顶点的有向圈。
Let D be a directed graph with vertex set V and order n. An anti-directed hamiltonian cycle H in D is a hamiltonian cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. An anti-directed 2-factor in D is a vertex-disjoint collection of anti-directed cycles in D that span V. It was proved in [3] that if the indegree and the outdegree of each vertex of D is greater than (9/16)n then D contains an anti-directed hamilton cycle. In this paper we prove that given a directed graph D, the problem of determining whether D has an anti-directed 2-factor is …
In this paper we generalise the even directed cycle problem, which asks whether a given digraph contains a directed cycle of even length, to orientations of regular matroids. We define non-even oriented matroids generalising non-even digraphs, which played a central role in resolving the computational complexity of the even dicycle problem. Then we show that the problem of detecting an even directed circuit in a regular matroid is polynomially equivalent to the recognition of non-even oriented matroids. Our main result is a precise characterisation of the class of non-even oriented bond matroi…
Hamilton Starters and Path Decompositions in Directed Circulants
针对q≥3、r≥1的有向循环图，构造了q层平衡哈密顿起点，在q=4（r≡1 mod4且r≥9）、q=3（r≡5 mod6且r足够大）时得到链兼容起点，实现弧集的最优路径分解。
Enumerating Hamiltonian Paths and Cycles
本文证明了完全多部图中定向哈密顿路径与回路的计数公式，采用折叠技术关联普通路径图与合法着色路径图。
In this paper we consider the question of the existence of Hamiltonian circuits in the tope graphs of central arrangements of hyperplanes. Some of the results describe connections between the existence of Hamiltonian circuits in the arrangement and the odd-even invariant of the arrangement. In conjunction with this, we present some results concerning bounds on the odd-even invariant. The results given here can be formulated more generally for oriented matroids and are still valid in that setting.
In his seminal paper from 1952 Dirac showed that the complete graph on $n\geq 3$ vertices remains Hamiltonian even if we allow an adversary to remove $\lfloor n/2\rfloor$ edges touching each vertex. In 1960 Ghouila-Houri obtained an analogue statement for digraphs by showing that every directed graph on $n\geq 3$ vertices with minimum in- and out-degree at least $n/2$ contains a directed Hamilton cycle. Both statements quantify the robustness of complete graphs (digraphs) with respect to the property of containing a Hamilton cycle. A natural way to generalize such results to arbitrary graphs (…
Circuit-partition of infinite matroids
本文为Komjáth等人关于有限匹配可扩展拟阵环划分的紧致性定理提供了简短证明，并推广了Thomassen猜想，证明了有限二元有向拟阵划分为有向环的充要条件。
A theorem of Cohn and Lempel [J. Combin. Theory Ser. A 13 (1972), 83-89] gives an equality relating the number of circuits in a directed circuit partition of a 2-in, 2-out digraph to the GF(2)-nullity of an associated matrix. This equality is essentially equivalent to the relationship between directed circuit partitions of 2-in, 2-out digraphs and vertex-nullity interlace polynomials of interlace graphs. We present an extension of the Cohn-Lempel equality that describes arbitrary circuit partitions in (undirected) 4-regular graphs. The extended equality incorporates topological results that ha…
We propose a new method for determining the elementary paths and elementary circuits in a directed graph. Also, the Hamiltonian paths and Hamiltonian circuits are enumerated.
Minimum degree edge-disjoint Hamilton cycles in random directed graphs
本文研究了二项随机有向图$D_{n,p}$中边不相交哈密顿圈的最大数量问题，证明了在特定条件下其数量恰好为最小出度与入度的较小值，并提供了随机化算法与新颖证明。
Bi-eulerian embeddings of graphs and digraphs
本文研究图与有向图的双欧拉嵌入，给出了可定向与不可定向双欧拉嵌入的存在条件，并将其与有向嵌入的最大亏格问题相联系，证明在特定度数条件下可构造由给定欧拉回路围成面的嵌入，同时隐含给出多项式时间算法。
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。
Hamilton decompositions of equal-side directed tori
本文研究了定向托里哈密顿分解问题，证明当d≥2且m为奇数时，D_d(m)的弧集可分解为d个定向哈密顿回路，结合多个定理和原理完成证明。
Hamilton decompositions of the directed 3-torus: a return-map and odometer view
本文证明了三维定向流形 D_3(m) 可分解为三个互不相交的定向哈密顿回路，通过分析返回映射和 odometer 结构，结合凯姆佩交换和符号不变量等方法，给出了构造证明。
We present an algebraic algorithm to detect the existence of and to list all indecomposable even circuits in a given graph. We also discuss an application of our work to the study of directed cycles in digraphs.
If eps > 0 and p >= n^{-1/2 + eps}, in a binomial random graph G(n,p) a.a.s. the set of cycles which can be constructed as a symmetric difference of Hamilton circuits is as large as parity by itself permits (all cycles if n is odd, all even cycles if n is even). Moreover, every p which ensures the above property a.a.s. must necessarily be such that for any constant c>0, eventually p >= (log n + 2 log log n + c)/n. So, whatever the smallest sufficient p for an a.a.s. Hamilton-generated cycle space might be, it does not coincide with the threshold for hamiltonicity of G(n,p).
We show that for each fixed $k$, the problem of finding $k$ pairwise vertex-disjoint directed paths between given source-sink pairs in a planar directed graph is solvable in polynomial time. In fact, it suffices to fix the number of faces needed to cover all sources and sinks. Moreover, the method can be extended to any fixed compact orientable surface (instead of the plane) and to rooted trees (instead of paths). Our approach is algebraic and is based on cohomology over graph (nonabelian) groups. More precisely, let $D=(V,A)$ be a directed graph and let $(G,\cdot)$ be a group. Call two functi…
The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveling salesman and other sequencing problems. We study the polytope by establishing its dimension, developing tools for the identification of facets, and using these tools to derive several families of facets. The tools include necessary and sufficient conditions for an inequality to be facet defining, and an algorithm for generating all undominated circuits. We use a novel approach to identifying families of facet-defining inequalities, based o…
Consider an electrical circuit $G$ each directed edge $e$ of which is a semiconductor with a monomial conductance function $y_e^* = f_e(y_e) = y_e^s / μ_e^r$ if $y_e \geq 0$ and $y_e^* = 0$ if $y_e \leq 0$. Here $e$ is a directed edge, $y_e$ is the potential difference (voltage), $y_e^*$ is the current in $e$, and $μ_e$ is the resistance of $e$; furthermore, $r$ and $s$ are two strictly positive real parameters common for all edges. In particular, case $r = s = 1$ corresponds to the Ohm law, while $r = \frac{1}{2}, s =1$ may be interpreted as the square law of resistance typical for hydraulics…
Hamilton decompositions of the directed 7-torus at odd modulus via root-flat certificates and a prefix-count construction
本文通过根平证书和前缀计数构造，证明了定向7维正方扭棱柱在奇整数m≥3时的哈密顿分解，解决了前缀计数方法无法处理m=3和m=5的情况。
The generating function that records the sizes of directed circuit partitions of a connected 2-in, 2-out digraph D can be determined from the interlacement graph of D with respect to a directed Euler circuit; the same is true of the generating functions for other kinds of circuit partitions. The interlace polynomials of Arratia, Bollobás and Sorkin [J. Combin. Theory Ser. B 92 (2004) 199-233; Combinatorica 24 (2004) 567-584] extend the corresponding functions from interlacement graphs to arbitrary graphs. We introduce a multivariate interlace polynomial that is an analogous extension of a mult…
Let $D$ be an oriented classical or virtual link diagram with directed universe $\vec{U}$. Let $C$ denote a set of directed Euler circuits, one in each connected component of $U$. There is then an associated looped interlacement graph $L(D,C)$ whose construction involves very little geometric information about the way $D$ is drawn in the plane; consequently $L(D,C)$ is different from other combinatorial structures associated with classical link diagrams, like the checkerboard graph, which can be difficult to extend to arbitrary virtual links. $L(D,C)$ is determined by three things: the structu…
Short Second Proof of the Odd-Modulus Directed Torus Hamilton Decomposition Theorem
本文针对奇数模有向环面图，提出一种基于固定行和选择定理的构造性证明，通过循环提升和纤维返回机制，将图分解为有向哈密顿环。
Hamilton decompositions of the directed 5-torus for odd modulus
本文证明了当 m 为奇数且 m≥3 时，定向五维 torus D_5(m) 存在 Hamilton 分解，通过构造方法和零集拉丁表证明了其分解性。
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。
In this article we discuss the question of presence of Hamiltonian cycle in the un-directed power graph of a group. In the process we develop weighted Hamiltonian cycle concept and prove a few general results regarding the Hamiltonian question.
Backward Arcs in Hamilton Oriented Cycles and Paths in Directed Graphs with Independence Number Two
本文研究了独立数为二的有向图中哈密顿有向环和路径的反向弧数量，证明了2连通图中最多有五个反向弧，1连通图中最多有两个反向弧。
Quantum circuit for exponentiation of Hamiltonians: an algorithmic description based on tensor products
本文针对哈密顿量指数化问题，提出一种基于张量积的算法方法，用于构造紧凑且易于实现的量子电路，以高效近似 e^(-i.t.U_1⊗U_2⊗…⊗U_n)。
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