Peter Keevash, Katherine Staden
We prove that any quasirandom dense large graph in which all degrees are equal and even can be decomposed into any given collection of two-factors (2-regular spanning subgraphs). A special case of this result gives a new solution to the Oberwolfach problem.
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The Directed Oberwolfach Problem can be considered as the directed version of the well-known Oberwolfach Problem, first mentioned by Ringel at a conference in Oberwolfach, Germany in 1967. In this paper, we describe some new partial results on the Directed Oberwolfach Problem with variable cycle lengths. In particular, we show that the complete symmetric digraph $K_n^{*}$ admits a $( \vec{C}_2, ..., \vec{C}_2, \vec{C}_3) $-factorization for all $ n\equiv 1, 3,$ or $ 7\pmod{8}$. We also show that $K_n^{*}$ admits a $(\vec{C}_2, \vec{C}_{n-2})$-factorization for any integer $n \geq 5$.
Various results on factorisations of complete graphs into circulant graphs and on 2-factorisations of these circulant graphs are proved. As a consequence, a number of new results on the Oberwolfach Problem are obtained. For example, a complete solution to the Oberwolfach Problem is given for every 2-regular graph of order 2p where p = 5 (mod 8) is prime.
Completing the solution of the directed Oberwolfach problem with cycles of equal length
本文通过证明完全对称有向图$K^*_{2m}$可分解为奇数长度的有向圈，彻底解决了桌长为奇数的双桌情形，从而完全解决了具有等长桌的有向Oberwolfach问题。
The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of $K_{2n+1}$ into edge-disjoint copies of a given $2$-factor. We show that this can be achieved for all large $n$. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large $n$.
A complete solution to the directed Oberwolfach problem of order $2 \pmod{4}$ with cycles of even lengths
本文解决了阶数为2 mod 4的有向奥伯沃尔夫ach问题，通过偶数长度循环的有向2-因子分解提供完整解法。
Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem, $OP(F)$, asks for a $2$-factorization of the complete graph on $v$ vertices in which each $2$-factor is isomorphic to $F$. In this paper, we give a complete solution to the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group $G$. We will also consider the same problem in the more general contest of graphs $F$ that are spanning subgraphs of an infinite complete graph $\mathbb{K}$ and we provide a solution when $F$ is locally f…
We show that the complete symmetric digraph $K_{2m}^\ast$ admits a resolvable decomposition into directed cycles of length $m$ for all odd $m$, $5 \le m \le 49$. Consequently, $K_{n}^\ast$ admits a resolvable decomposition into directed cycles of length $m$ for all $n \equiv 0 \pmod{2m}$ and odd $m$, $5 \le m \le 49$.
Completing the solution of the directed Oberwolfach problem with two tables
本文解决了两个不同长度桌子的有向Oberwolfach问题的最后一个未解情形，证明了$K^*_n$可分解为两个长度分别为$t_1\in\{4,6\}$和偶数$t_2$的不相交有向环，且$t_1+t_2\geqslant14$，结合已有结果给出完整解。
The Oberwolfach Problem $OP(F)$ -- posed by Gerhard Ringel in 1967 -- is a paradigmatic Combinatorial Design problem asking whether the complete graph $K_v$ decomposes into edge-disjoint copies of a $2$-regular graph $F$ of order $v$. In this paper, we provide all the necessary equipment to generate solutions to $OP(F)$ for relatively small orders by using the so-called difference methods. From the theoretical standpoint, we present new insights on the combinatorial structures involved in the solution of the problem. Computationally, we provide a full recipe whose base ingredients are advanced…
We examine the necessary and sufficient conditions for a complete symmetric equipartite digraph $K_{n[m]}^\ast$ with $n$ parts of size $m$ to admit a resolvable decomposition into directed cycles of length $t$. We show that the obvious necessary conditions are sufficient for $m,n,t \ge 2$ in each of the following four cases: (i) $m(n-1)$ is even; (ii) $\gcd(m,n) \not\in \{1,3\}$; (iii) $\gcd(m,n)=1$ and $4|n$ or $6|n$; and (iv) $\gcd(m,n)=3$, and if $n=6$, then $p|m$ for a prime $p \le 37$.
A constructive solution to the Oberwolfach Problem with a large cycle
本文针对未解决的 Oberwolfach 问题，提出了一种构造性解法，利用合并-分离技术与近正则自同构群方法，解决了当 $F$ 包含长度大于显式下界的圈时的 $2$-因子分解问题。
A survey on constructive methods for the Oberwolfach problem and its variants
本文综述了Oberwolfach问题及其变体的构造性方法，重点介绍放大构造、二部因子情形、循环图方法及具有良好自同构性质的解。
A complete solution to the generalized honeymoon Oberwolfach problem with one round table
本文完全解决了带一个圆桌的广义蜜月Oberwolfach问题，证明了其明显必要条件也是充分的。
On the spouse-loving variant of the Oberwolfach problem
本文研究 Oberwolfach 问题的爱配偶变体，证明偶数阶完全图加重复 1-因子可分解为指定长度圈的 2-因子，并给出主要充要条件及全偶长度情形的分解结果。
The concept of a $1$-rotational factorization of a complete graph under a finite group $G$ was studied in detail by Buratti and Rinaldi. They found that if $G$ admits a $1$-rotational $2$-factorization, then the involutions of $G$ are pairwise conjugate. We extend their result by showing that if a finite group $G$ admits a $1$-rotational $k=2^nm$-factorization where $n\geq 1$, and $m$ is odd, then $G$ has at most $m(2^n-1)$ conjugacy classes containing involutions. Also, we show that if $G$ has exactly $m(2^n-1)$ conjugacy classes containing involutions, then the product of a central involutio…
On the Oberwolfach problem for single-flip $2$-factors via graceful labelings
本文利用优美标号和加倍构造方法，证明了具有单翻转自同构且包含足够大圈的 $2$-正则图的 Oberwolfach 问题及其覆盖与装填版本可解，并显式构造了解。
The Honeymoon Oberwolfach Problem: small cases
本文研究蜜月奥伯沃尔法赫问题，通过计算机搜索将已有结论从 n≤9 推广到所有 n≤20 的小参数情形，证明其满足必要条件时必有解。
On the Generalized Honeymoon Oberwolfach Problem
本文研究了具有多个圆桌的广义霍纳顿奥伯沃尔夫ach问题，证明了在特定条件下该问题存在解。
The directed Oberwolfach problem with variable cycle lengths: a recursive construction
本文针对变长循环的有向 Oberwolfach 问题提出递归构造方法，利用较小规模的解扩展构造更大规模的解，并据此解决双表情形及总长不超过13的多种情形。
On the minisymposium problem
本文提出与广义Oberwolfach问题相关的minisymposium问题，利用Hilton和Johnson的方法给出部分情形的解，并在环长均为k且m整除v时得到大量结果。
The generalization of the Oberwolfach Problem, proposed by J. Liu in 2000, asks for a uniform $2$-factorization of the complete multipartite graph $K_{m\times n}$. Here we focus our attention on $2$-factorizations regular under the cyclic group $Z_{mn}$, whose $2$-factors are disjoint union of cycles all of even length $\ell$. In particular, we present a complete solution for the extremal cases $\ell=4$ and $\ell=mn$.
We describe a method that we believe may be foundational for a comprehensive theory of generalised Turan problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that any 3-graph on n vertices for which every pair of vertices is contained in more than n/2 edges must contain a …
Let $\mathcal G$ be a separable family of graphs. Then for all positive constants $ε$ and $Δ$ and for every sufficiently large integer $n$, every sequence $G_1,\dotsc,G_t\in\mathcal G$ of graphs of order $n$ and maximum degree at most $Δ$ such that $e(G_1)+\dotsb+e(G_t) \leq (1-ε)\binom{n}{2}$ packs into $K_n$. This improves results of Böttcher, Hladký, Piguet, and Taraz when $\mathcal G$ is the class of trees and of Messuti, Rödl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of…
The problem of existence of closed knight's tours in $[n]^d$, where $[n]=\{0, 1, \dots, n-1\}$, was recently solved by Erde, Golénia, and Golénia. They raised the same question for a generalised, $(a, b)$ knight, which is allowed to move along any two axes of $[n]^d$ by $a$ and $b$ unit lengths respectively. Given an even number $a$, we show that the $[n]^d$ grid admits an $(a, 1)$ knight's tour for sufficiently large even side length $n$.
A generalised Ramsey--Turán problem for matchings
本文证明了关于匹配的广义 Ramsey--Turán 定理，通过显式构造和基于 Gallai--Edmonds 分解的复杂压缩算法，确定了在特定边着色条件下图中完全子图的最大数量。
Ramsey-type problems for generalised Sidon sets
受Erdős等学者相关问题启发，研究者证明了具备额外Ramsey型性质的广义Sidon集是存在的，推进了相关组合数论问题的研究。
Perfect tilings of 3-graphs with the generalised triangle
本文确定了3-一致超图被广义三角形完美铺砌的最佳最小共度条件，并给出了彩虹版本的渐近最优条件。
The purpose of this note is to draw attention to problems related to a concept called majority colouring recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised a problem of determining, for a natural number $k$, the smallest number $m=m(k)$ such that every digraph can be coloured with $m$ colours where each vertex has the same colour as at most $1/k$ proportion of its out-neighbours. We show that $m(k)\in\{2k-1,2k\}$. We also prove a result supporting the conjecture that $m(2)=3$. Moreover, we prove similar results for a more general concept called majority choosabilit…
Given an edge-coloured graph, we say that a subgraph is rainbow if all of its edges have different colours. Let $\operatorname{ex}(n,H,$rainbow-$F)$ denote the maximal number of copies of $H$ that a properly edge-coloured graph on $n$ vertices can contain if it has no rainbow subgraph isomorphic to $F$. We determine the order of magnitude of $\operatorname{ex}(n,C_s,$rainbow-$C_t)$ for all $s,t$ with $s\not =3$. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer by showing that $\operatorname{ex}(n,C_{2k},$rainbow-$C_{2k})$ is $Θ(n^{k-1})$ if $k\geq 3$ and $Θ(n^2)$ if…
We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to longstanding decomposition problems. For instance, our results imply the following. Let $G$ be a quasi-random $n$-vertex graph and suppose $H_1,\dots,H_s$ are bounded degree $n$-vertex graphs with $\sum_{i=1}^{s} e(H_i) \leq (1-o(1)) e(G)$. Then $H_1,\dots,H_s$ can be packed edge-disjointly into $G$. The case when $G$ is the complete graph $K_n$ implies an approximate version of the tree packing conjecture of Gyárfás and Lehel for bounded degree trees, and of the Oberwolfach p…
The generalised Sudoku problem with $N$ symbols is known to be NP-complete, and hence is equivalent to any other NP-complete problem, even for the standard restricted version where $N$ is a perfect square. In particular, generalised Sudoku is equivalent to the, classical, Hamiltonian cycle problem. A constructive algorithm is given that reduces generalised Sudoku to the Hamiltonian cycle problem, where the resultant instance of Hamiltonian cycle problem is sparse, and has $O(N^3)$ vertices. The Hamiltonian cycle problem instance so constructed is a directed graph, and so a (known) conversion t…
We discuss recent progress on the problem of classifying point-primitive generalised polygons. In the case of generalised hexagons and generalised octagons, this has reduced the problem to primitive actions of almost simple groups of Lie type. To illustrate how the natural geometry of these groups may be used in this study, we show that if $\mathcal{S}$ is a finite thick generalised hexagon or octagon with $G \leqslant{\rm Aut}(\mathcal{S})$ acting point-primitively and the socle of $G$ isomorphic to ${\rm PSL}_n(q)$ where $n \geqslant 2$, then the stabiliser of a point acts irreducibly on the…
The 1-Steiner tree problem, the problem of constructing a Steiner minimum tree containing at most one Steiner point, has been solved in the Euclidean plane by Georgakopoulos and Papadimitriou using plane subdivisions called oriented Dirichlet cell partitions. Their algorithm produces an optimal solution within $O(n^2)$ time. In this paper we generalise their approach in order to solve the $k$-Steiner tree problem, in which the Steiner minimum tree may contain up to $k$ Steiner points for a given constant $k$. We also extend their approach further to encompass other normed planes, and to solve …
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