Asaf Ferber, Kaarel Haenni, Vishesh Jain
Let $H_1,\dots,H_k$ be Hamilton cycles in $K_n$, chosen independently and uniformly at random. We show, for $k = o(n^{1/100})$, that the probability of $H_1,\dots,H_k$ being edge-disjoint is $(1+o(1))e^{-2\binom{k}{2}}$. This extends a corresponding estimate obtained by Robbins in the case $k=2$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every α> 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + α)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree δmust have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every α> 0, every (almost) regular and sufficiently large graph on n vertices wit…
We show that provided $\log^{50} n/n \leq p \leq 1 - n^{-1/4}\log^9 n$ we can with high probability find a collection of $\lfloor δ(G)/2 \rfloor$ edge-disjoint Hamilton cycles in $G \sim G_{n, p}$, plus an additional edge-disjoint matching of size $\lfloor n/2 \rfloor$ if $δ(G)$ is odd. This confirms, for the above range of $p$, a conjecture of Frieze and Krivelevich.
Thomassen conjectured that there is a function $f(k)$ such that every strongly $f(k)$-connected tournament contains $k$ edge-disjoint Hamiltonian cycles. This conjecture was recently proved by Kühn, Lapinskas, Osthus, and Patel who showed that $f(k)\leq O(k^2(\log k)^2)$ and conjectured that there is a constant $C$ such that $f(k)\leq Ck^2$. We prove this conjecture.
Edge disjoint Hamilton cycles in random digraphs of constant minimum degree
研究随机图中在给定最小入度和出度条件下存在方向哈密顿回路的问题，证明在特定条件下图中存在k个边不相交的哈密顿回路。
Disjointness Graphs of segments in $R^2$ are almost all Hamiltonian
本文完整刻画了 $R^2$ 中点集的边不相交图的哈密顿性，证明仅有 8 个实例非哈密顿图，并据此推导出反例反驳了 Plotnikov 于 1998 年提出的哈密顿回路存在性准则。
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。
Edge-disjoint Hamilton cycles under a bipartite-hole condition
研究在二分洞条件下，图中边不相交哈密顿圈的存在情况。通过引入函数\(f(a,k)\)，利用删除引理、哈密顿性定理及贪心填充论证证明其上下界，即\(f(a,k)=\Theta\left(a+k+\frac{ak}{\log(k+2)}\right)\)。
We prove that any one-ended, locally finite Cayley graph with non-torsion generators admits a decomposition into edge-disjoint Hamiltonian (i.e. spanning) double-rays. In particular, the $n$-dimensional grid $\mathbb{Z}^n$ admits a decomposition into $n$ edge-disjoint Hamiltonian double-rays for all $n \in \mathbb{N}$.
In a recent paper, we showed that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. The main consequence of this theorem is that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large. This verified a conjecture of Kelly from 1968. In this paper, we derive a number of further consequences of our result on robust outexpanders, the main ones are the following: (i) an undirected analogue of our r…
The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected $2k$-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into $k$ edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every $2k$-valent connected circulant graph has a decomposition into $k$ edge-disjoint Hamilton cycles. We settle the problem of decomposing $2k$-valent infinite cir…
We show that if $C_1$ and $C_2$ are directed cycles (of length at least two), then the Cartesian product $C_1 \Box C_2$ has two arc-disjoint hamiltonian paths. (This answers a question asked by J. A. Gallian in 1985.) The same conclusion also holds for the Cartesian product of any four or more directed cycles (of length at least two), but some cases remain open for the Cartesian product of three directed cycles. We also discuss the existence of arc-disjoint hamiltonian paths in $2$-generated Cayley digraphs on (finite or infinite) abelian groups.
The spined cube $SQ_n$ is a variant of the hypercube $Q_n$, introduced by Zhou et al. in [Information Processing Letters 111 (2011) 561-567] as an interconnection network for parallel computing. A graph $\G$ is an $m$-Cayley graph if its automorphism group $\Aut(\G)$ has a semiregular subgroup acting on the vertex set with $m$ orbits, and is a Caley graph if it is a 1-Cayley graph. It is well-known that $Q_n$ is a Cayley graph of an elementary abelian 2-group $\mz_2^n$ of order $2^n$. In this paper, we prove that $SQ_n$ is a 4-Cayley graph of $\mz_2^{n-2}$ when $n\geq6$, and is a $\lfloor n/2\…
We prove a conjecture of Penrose about the standard random geometric graph process, in which n vertices are placed at random on the unit square and edges are sequentially added in increasing order of lengths taken in the l_p norm. We show that the first edge that makes the random geometric graph Hamiltonian is a.a.s. exactly the same one that gives 2-connectivity. We also extend this result to arbitrary connectivity, by proving that the first edge in the process that creates a k-connected graph coincides a.a.s. with the first edge that causes the graph to contain k/2 pairwise edge-disjoint Ham…
We present a tight extremal threshold for the existence of Hamilton cycles in graphs with large minimum degree and without a large ``bipartite hole`` (two disjoint sets of vertices with no edges between them). This result extends Dirac's classical theorem, and is related to a theorem of Chvátal and Erdős. In detail, an $(s, t)$-bipartite-hole in a graph $G$ consists of two disjoint sets of vertices $S$ and $T$ with $|S|= s$ and $|T|=t$ such that there are no edges between $S$ and $T$; and $\widetildeα(G)$ is the maximum integer $r$ such that $G$ contains an $(s, t)$-bipartite-hole for every pa…
Consider the random subgraph process on a base graph $G$ on $n$ vertices: a sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of random subgraphs of $G$ obtained by choosing an ordering of the edges of $G$ uniformly at random, and by sequentially adding edges to $G_0$, the empty graph on the vertex set of $G$, according to the chosen ordering. We show that if $G$ has one of the following properties: 1. There is a positive constant $\varepsilon > 0$ such that $δ(G) \geq \left( \frac{1}{2} + \varepsilon \right) n$; 2. There are some constants $α, β>0$ such that every two disjoint subsets $U,W$ of …
We show that every sufficiently large regular tournament can almost completely be decomposed into edge-disjoint Hamilton cycles. More precisely, for each η>0 every regular tournament G of sufficiently large order n contains at least (1/2-η)n edge-disjoint Hamilton cycles. This gives an approximate solution to a conjecture of Kelly from 1968. Our result also extends to almost regular tournaments.
Consider the random graph process $\{G_t\}_{t\geq 0}$. For $k\geq 3$ let $G_{t}^{(k)}$ denote the $k$-core of $G_t$ and let $τ_k$ be the minimum $t$ such that the $k$-core of $G_t$ is nonempty. It is well known that w.h.p. for $G_{τ_k}^{(k)}$ has linear size while it is believed to be Hamiltonian. Bollobás, Cooper, Fenner and Frieze further conjectured that w.h.p. $G_{t}^{(k)}$ spans $\lfloor \frac{k-1}{2} \rfloor$ edge-disjoint Hamilton cycles plus, when $k$ is even, a perfect matching for $t\geq τ_k$. We prove that w.h.p.\@ if $k$ is odd then $G_{t}^{(k)}$ spans $\frac{k-3}{2}$ edge disjoint…
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…
We consider hypercubes with pairwise disjoint faulty edges. An $n$-dimensional hypercube $Q_n$ is an undirected graph with $2^n$ nodes, each labeled with a distinct binary strings of length $n$. The parity of the vertex is 0 if the number of ones in its labels is even, and is 1 if the number of ones is odd. Two vertices $a$ and $b$ are connected by the edge iff $a$ and $b$ differ in one position. If $a$ and $b$ differ in position $i$, then we say that the edge $(a,b)$ goes in direction $i$ and we define the parity of the edge as the parity of the end with 0 on the position $i$. It was already …
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…
We prove that there exists a positive constant εsuch that if \log n / n \le p \le n^{-1+ε}, then asymptotically almost surely the random graph G ~ G(n,p) contains a collection of \lfloor δ(G)/2 \rfloor edge-disjoint Hamilton cycles.
We prove that a complete multipartite graph $K$ with $n>1$ vertices and $m$ edges can be decomposed into edge-disjoint Hamilton paths if and only if $\frac m{n-1}$ is an integer and the maximum degree of $K$ is at most $\frac {2m}{n-1}$.
We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree d = (1/2+a)n. For any constant a > 0, we give an optimal answer in the following sense: let reg_even(n,d) denote the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree d. Then the number of edge-disjoint Hamilton cycles we find equals reg_even(n,d)/2. The value of reg_even(n,d) is known for infinitely many values of n and d. We also extend our results to graphs G of minimum degree d >= n/2, unle…
We say that a $k$-uniform hypergraph $C$ is a Hamilton cycle of type $\ell$, for some $1\le \ell \le k$, if there exists a cyclic ordering of the vertices of $C$ such that every edge consists of $k$ consecutive vertices and for every pair of consecutive edges $E_{i-1},E_i$ in $C$ (in the natural ordering of the edges) we have $|E_{i-1}-E_i|=\ell$. We prove that for $\ell \le k\le 2\ell$, with high probability almost all edges of a random $k$-uniform hypergraph $H(n,p,k)$ with $p(n)\gg \log^2 n/n$ can be decomposed into edge disjoint type $\ell$ Hamilton cycles. We also provide sufficient condi…
Let $H$ be an $n$-vertex 3-uniform hypergraph such that every pair of vertices is in at least $n/3+o(n)$ edges. We show that $H$ contains two vertex-disjoint tight paths whose union covers the vertex set of $H$. The quantity two here is best possible and the degree condition is asymptotically best possible. This result also has an interpretation as the \emph{deficiency problems}, recently introduced by Nenadov, Sudakov and Wagner: every such $H$ can be made Hamiltonian by adding at most two vertices and all triples intersecting them.
We show that if pn >> log n, the binomial random graph G_{n,p} has an approximate Hamilton decomposition. More precisely, we show that in this range G_{n,p} contains a set of edge-disjoint Hamilton cycles covering almost all of its edges. This is best possible in the sense that the condition that pn >> log n is necessary.
Extremal problems on disjoint path covers of graphs
本文研究图的不相交路径覆盖的极值问题，通过推广哈密顿性结果，分别依据团数量与\(\alpha\)-谱半径，建立图具有不同类型\(t\)不相交路径覆盖的充分条件，并刻画达到这些界的极值图。
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.
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…
For a graph $G$ the random $n$-lift of $G$ is obtained by replacing each of its vertices by a set of $n$ vertices, and joining a pair of sets by a random matching whenever the corresponding vertices of $G$ are adjacent. We show that asymptotically almost surely the random lift of a graph $G$ is hamiltonian, provided $G$ has the minimum degree at least $5$ and contains two disjoint Hamiltonian cycles whose union is not a bipartite graph.
In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on only two properties: one requiring expansion of ``small'' sets, the other ensuring the existence of an edge between any two disjoint ``large'' sets. We also discuss applications in positional games, random graphs and extremal graph theory.
Bispanning graphs are undirected graphs with an edge set that can be decomposed into two disjoint spanning trees. The operation of symmetrically swapping two edges between the trees, such that the result is a different pair of disjoint spanning trees, is called an edge exchange or a symmetric base exchange. The graph of symmetric base exchanges of a bispanning graph contains a vertex for every valid pair of disjoint spanning trees, and edges between them to represent all possible edge exchanges. We are interested in a restriction of these graphs to only unique symmetric base exchanges, which a…
We study Hamiltonicity in random subgraphs of the hypercube $\mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $\mathcal{Q}^n$ according to a uniformly chosen random ordering. Then, with high probability, as soon as the graph produced by this process has minimum degree $2k$, it contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. Secondly, we obtain a perturbation result: if $H\subseteq\mathcal{Q}^n$ satisfies $δ(H)\geqαn$ with $α>0$ fixed and we consider a random binomial subgraph $\mathcal{…
A Hamilton decomposition of a graph is a partitioning of its edge set into disjoint spanning cycles. The existence of such decompositions is known for all hypercubes of even dimension $2n$. We give a decomposition for the case $n = 2^a3^b$ that is highly symmetric in the sense that every cycle can be derived from every other cycle just by permuting the axes. We conjecture that a similar decomposition exists for every n.
In this paper we study the randomly edge colored graph that is obtained by adding randomly colored random edges to an arbitrary randomly edge colored dense graph. In particular we ask how many colors and how many random edges are needed so that the resultant graph contains a fixed number of edge disjoint rainbow Hamilton cycles. We also ask when in the resultant graph every pair of vertices is connected by a rainbow path.
We give a description of the cycle structure of the Heawood graph, $C_{14}$. In particular, we prove that the automorphism group of $C_{14}$ acts transitively on the set of $12$-cycles, Hamiltonian cycles, and disjoint pairs of $6$-cycles. We also enumerate $12$-, $10$-, and $8$-cycles in $C_{14}$, as well as pairs of disjoint $6$-cycles.
Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no two cycles share a common independent set of size more than $k$. We shall mainly be interested in the behavior of $f(n,k)$ when $k$ is a linear function of $n$, namely $k=cn$. We show a threshold phenomenon: there exists a constant $c_t$ such that for $c<c_t$, $f(n,cn)$ is bounded by a constant depending only on $c$ and not on $n$, and for $c_t <c$, $f(n,cn)…
Paired 2-disjoint path covers of Bcube under the partitioned edge fault model
本文研究了在分区边故障模型下BCube的容错配对2不相交路径覆盖问题，证明了在大量边故障情况下BCube仍能保持配对2-DPC的鲁棒性。
Isaak posed the following problem. Suppose $T$ is a tournament having a minimum feedback arc set which induces an acyclic digraph with a hamiltonian path. Is it true that the maximum number of arc-disjoint cycles in $T$ equals the cardinality of minimum feedback arc set of $T$? We prove that the answer to the problem is in the negative. Further, we study the number of arc-disjoint cycles through a vertex $v$ of the minimum out-degree in an oriented graph $D$. We prove that if $v$ is adjacent to all other vertices, then $v$ belongs to $δ^+(D)$ arc-disjoint cycles.
In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $χ'(G)=D$. (ii) [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 \rfloor $. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) We prove an optimal result on the number of edge-disjoint …
It is known that w.h.p. the hitting time $τ_{2σ}$ for the random graph process to have minimum degree $2σ$ coincides with the hitting time for $σ$ edge disjoint Hamilton cycles. In this paper we prove an online version of this property. We show that, for a fixed integer $σ\geq 2$, if random edges of $K_n$ are presented one by one then w.h.p. it is possible to color the edges online with $σ$ colors so that at time $τ_{2σ}$, each color class is Hamiltonian.
Let $n_{1}$ and $n_{2}$ be two integers with $n_{1},n_{2}\geq3$ and $G$ a graph of order $n=n_{1}+n_{2}$. As a generalization of Ore's degree condition for the existence of Hamilton cycle in $G$, El-Zahar proved that if $δ(G)\geq \left\lceil\frac{n_{1}}{2}\right\rceil+\left\lceil\frac{n_{2}}{2}\right\rceil$ then $G$ contains two disjoint cycles of length $n_{1}$ and $n_{2}$. Recently, Yan et. al considered the problem by extending the degree condition to degree sum condition and proved that if $d(u)+d(v)\geq n+4$ for any pair of non-adjacent vertices $u$ and $v$ of $G$, then $G$ contains two d…
The Pairing-Hamiltonian property in Cartesian products of graphs
本文证明图的配对哈密顿性质在笛卡尔积下保持，并给出充要条件，指出无支撑限制时逆命题不成立。
$s$-Stable Kneser Graph are Hamiltonian
本文研究 $s$-稳定 Kneser 图的结构性质，并证明所有连通的此类图都具有哈密顿回路。
Paired many-to-many 2-disjoint path cover of Johnson graphs
本文研究 Johnson 图的两两多对多 2-不相交路径覆盖问题，证明 Johnson 图 $J(n,k)$ 及其衍生图族 $QJ(n,k)$ 均为两两 2-可覆盖的，从而加强了其 Hamilton 连通性结论。
The problem of packing Hamilton cycles in random and pseudorandom graphs has been studied extensively. In this paper, we look at the dual question of covering all edges of a graph by Hamilton cycles and prove that if a graph with maximum degree $Δ$ satisfies some basic expansion properties and contains a family of $(1-o(1))Δ/2$ edge disjoint Hamilton cycles, then there also exists a covering of its edges by $(1+o(1))Δ/2$ Hamilton cycles. This implies that for every $α>0$ and every $p \geq n^{α-1}$ there exists a covering of all edges of $G(n,p)$ by $(1+o(1))np/2$ Hamilton cycles asymptotically…
In 1981 Jackson showed that the diregular bipartite tournament (a complete bipartite graph whose edges are oriented so that every vertex has the same in- and outdegree) contains a Hamilton cycle, and conjectured that in fact the edge set of it can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: For every $c>1/2$ and $\varepsilon>0$ there exists $n_0$ such that every $cn$-regular bipartite digraph on $2n\geq n_0$ vertices contains $(1-\varepsilon)cn$ edge-disjoint Hamilton cycles.
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 $p \geq \log ^{1 + o(1)}n/n$.
Extensions of Erdős's 1962 theorem on non-Hamiltonian graphs
本文基于Erdős 1962年定理，扩展了非哈密顿图的极值理论，解决了多个开放问题并改进了相关结果。
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…
Let $k$, $λ$ and $μ$ be positive integers. A decomposition of a multigraph $ λG$ into edge-disjoint subgraphs $G_1, \ldots , G_k$ is said to be \emph{enclosed} by a decomposition of a multigraph $μH$ into edge-disjoint subgraphs $H_1, \ldots , H_k$ if $μ> λ$ and $G_i$ is a subgraph of $H_i$, $1 \leq i \leq k$. In this paper we initiate the study of when a decomposition can be enclosed by a decomposition that consists of spanning subgraphs. A decomposition of a graph is a 2-factorization if each subgraph is 2-regular and is Hamiltonian if each subgraph is a Hamiltonian cycle. Let $n$ and $m$ be…
A Hamilton cycle in a graph $Γ$ is a cycle passing through every vertex of $Γ$. A Hamiltonian decomposition of $Γ$ is a partition of its edge set into disjoint Hamilton cycles. One of the oldest results in graph theory is Walecki's theorem from the 19th century, showing that a complete graph $K_n$ on an odd number of vertices $n$ has a Hamiltonian decomposition. This result was recently greatly extended by Kühn and Osthus. They proved that every $r$-regular $n$-vertex graph $Γ$ with even degree $r=cn$ for some fixed $c>1/2$ has a Hamiltonian decomposition, provided $n=n(c)$ is sufficiently lar…
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…
A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in G_n,p a.a.s. has size \lfloor delta(G_n,p) /2 \rfloor. Glebov, Krivelevich and Szabó recently initiated research on the `dual' problem, where one asks for a set of Hamilton cycles covering all edges of G. Our main result states that for log^{117}n / n < p < 1-n^{-1/8}, a.a.s. the edges of G_n,p can be covered by \lceil Delta(G_n,p)/2 \rceil Hamilton cycles. This is clearly optimal and improv…
An edge (vertex) cut $X$ of $G$ is $r$-essential if $G-X$ has two components each of which has at least $r$ edges. A graph $G$ is $r$-essentially $k$-edge-connected (resp. $k$-connected) if it has no $r$-essential edge (resp. vertex) cuts of size less than $k$. If $r=1$, we simply call it essential. Recently, Lai and Li proved that every $m$-edge-connected essentially $h$-edge-connected graph contains $k$ edge-disjoint spanning trees, where $k,m,h$ are positive integers such that $k+1\le m\le 2k-1$ and $h\ge \frac{m^2}{m-k}-2$. In this paper, we show that every $m$-edge-connected and $2$-essen…
Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C \subset H is a collection of N edges for which there is an ordering of the vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i, v_{i+1}, v_{i+2}} is an edge of C (indices are considered modulo N). We develop new techniques which enable us to prove that under certain natural pseudo-random conditions, almost all edges of H can be covered by edge-disjoint tight Hamilton cycles, for N divisible by 4. Consequently, we derive the corollary that random 3-uniform hypergraphs can be almost completely packed…
A long-standing conjecture of Kelly states that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove this conjecture for large n. In fact, we prove a far more general result, based on our recent concept of robust expansion and a new method for decomposing graphs. We show that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. This enables us to obtain numerous further results, e.g. as a special case we confirm a con…
We give a new proof of the Skeletal Lemma, which is the main technical tool in our paper on Hamilton cycles in line graphs [T. Kaiser and P. Vrána, Hamilton cycles in 5-connected line graphs, European J. Combin. 33 (2012), 924-947]. It generalises results on disjoint spanning trees in graphs to the context of 3-hypergraphs. The lemma is proved in a slightly stronger version that is more suitable for applications. The proof is simplified and formulated in a more accessible way.
A famous result by Rödl, Ruciński, and Szemerédi guarantees a (tight) Hamilton cycle in $k$-uniform hypergraphs $H$ on $n$ vertices with minimum $(k-1)$-degree $δ_{k-1}(H)\geq (1/2+o(1))n$, thereby extending Dirac's result from graphs to hypergraphs. For graphs, much more is known; each graph on $n$ vertices with $δ(G)\geq (1/2+o(1))n$ contains $(1-o(1))r$ edge-disjoint Hamilton cycles where $r$ is the largest integer such that $G$ contains a spanning $2r$-regular subgraph, which is clearly asymptotically optimal. This was proved by Ferber, Krivelevich, and Sudakov answering a question raised …
Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. Determining toughness is an NP-hard problem for arbitrary graphs. The Toughness Conjecture of Chvátal, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is hamiltonian, is still open in general. A graph is called $(P_2\cup P_3)$-free if it does not contain any induced subgraph isomorphic to $P_2\cup P_3$, the union of two vertex-disjoint paths of order 2 and 3, resp…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。