Infinitely many cyclic solutions to the Hamilton-Waterloo problem with odd length cycles

It is conjectured that for every pair $(\ell,m)$ of odd integers greater than 2 with $m \equiv 1\; \pmod{\ell}$, there exists a cyclic two-factorization of $K_{\ell m}$ having exactly $(m-1)/2$ factors of type $\ell^m$ and all the others of type $m^{\ell}$. The authors prove the conjecture in the affirmative when $\ell \equiv 1\; \pmod{4}$ and $m \geq \ell^2 -\ell + 1$.Comment: 31 page

A constructive solution to the Oberwolfach Problem with a large cycle

For every $2$-regular graph $F$ of order $v$, the Oberwolfach problem $OP(F)$ asks whether there is a $2$-factorization of $K_v$ ($v$ odd) or $K_v$ minus a $1$-factor ($v$ even) into copies of $F$. Posed by Ringel in 1967 and extensively studied ever since, this problem is still open. In this paper we construct solutions to $OP(F)$ whenever $F$ contains a cycle of length greater than an explicit lower bound. Our constructions combine the amalgamation-detachment technique with methods aimed at building $2$-factorizations with an automorphism group having a nearly-regular action on the vertex-set.Comment: 11 page

On the full automorphism group of a Hamiltonian cycle system of odd order

It is shown that a necessary condition for an abstract group G to be the full automorphism group of a Hamiltonian cycle system is that G has odd order or it is either binary, or the affine linear group AGL(1; p) with p prime. We show that this condition is also sufficient except possibly for the class of non-solvable binary groups.Comment: 11 pages, 2 figure

Vertex-regular 11-factorizations in infinite graphs

The existence of $1$-factorizations of an infinite complete equipartite graph $K_m[n]$ (with $m$ parts of size $n$) admitting a vertex-regular automorphism group $G$ is known only when $n=1$ and $m$ is countable (that is, for countable complete graphs) and, in addition, $G$ is a finitely generated abelian group $G$ of order $m$. In this paper, we show that a vertex-regular $1$-factorization of $K_m[n]$ under the group $G$ exists if and only if $G$ has a subgroup $H$ of order $n$ whose index in $G$ is $m$. Furthermore, we provide a sufficient condition for an infinite Cayley graph to have a regular $1$-factorization. Finally, we construct 1-factorizations that contain a given subfactorization, both having a vertex-regular automorphism group

A survey on constructive methods for the Oberwolfach problem and its variants

The generalized Oberwolfach problem asks for a decomposition of a graph $G$ into specified 2-regular spanning subgraphs $F_1,\ldots, F_k$, called factors. The classic Oberwolfach problem corresponds to the case when all of the factors are pairwise isomorphic, and $G$ is the complete graph of odd order or the complete graph of even order with the edges of a $1$-factor removed. When there are two possible factor types, it is called the Hamilton-Waterloo problem. In this paper we present a survey of constructive methods which have allowed recent progress in this area. Specifically, we consider blow-up type constructions, particularly as applied to the case when each factor consists of cycles of the same length. We consider the case when the factors are all bipartite (and hence consist of even cycles) and a method for using circulant graphs to find solutions. We also consider constructions which yield solutions with well-behaved automorphisms.Comment: To be published in the Fields Institute Communications book series. 23 pages, 2 figure

On the minisymposium problem

The generalized Oberwolfach problem asks for a factorization of the complete graph $K_v$ into prescribed $2$-factors and at most a $1$-factor. When all $2$-factors are pairwise isomorphic and $v$ is odd, we have the classic Oberwolfach problem, which was originally stated as a seating problem: given $v$ attendees at a conference with $t$ circular tables such that the $i$th table seats $a_i$ people and ${\sum_{i=1}^t a_i = v}$, find a seating arrangement over the $\frac{v-1}{2}$ days of the conference, so that every person sits next to each other person exactly once. In this paper we introduce the related {\em minisymposium problem}, which requires a solution to the generalized Oberwolfach problem on $v$ vertices that contains a subsystem on $m$ vertices. That is, the decomposition restricted to the required $m$ vertices is a solution to the generalized Oberwolfach problem on $m$ vertices. In the seating context above, the larger conference contains a minisymposium of $m$ participants, and we also require that pairs of these $m$ participants be seated next to each other for $\left\lfloor\frac{m-1}{2}\right\rfloor$ of the days. When the cycles are as long as possible, i.e.\ $v$, $m$ and $v-m$, a flexible method of Hilton and Johnson provides a solution. We use this result to provide further solutions when $v \equiv m \equiv 2 \pmod 4$ and all cycle lengths are even. In addition, we provide extensive results in the case where all cycle lengths are equal to $k$, solving all cases when $m\mid v$, except possibly when $k$ is odd and $v$ is even.Comment: 25 page

Merging Combinatorial Design and Optimization: the Oberwolfach Problem

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 Combinatorial Design Theory, so-called difference methods represent a well-known solution technique and construct solutions in infinitely many cases exploiting symmetric and balanced structures. This approach reduces the problem to finding a well-structured $2$-factor which allows us to build solutions that we call $1$- or $2$-rotational according to their symmetries. We tackle $OP$ by modeling difference methods with Optimization tools, specifically Constraint Programming ($CP$) and Integer Programming ($IP$), and correspondingly solve instances with up to $v=120$ within $60s$. In particular, we model the $2$-rotational method by solving in cascade two subproblems, namely the binary and group labeling, respectively. A polynomial-time algorithm solves the binary labeling, while $CP$ tackles the group labeling. Furthermore, we prov ide necessary conditions for the existence of some $1$-rotational solutions which stem from computational results. This paper shows thereby that both theoretical and empirical results may arise from the interaction between Combinatorial Design Theory and Operation Research

Armida. Excerpts - Don Mus.Ms. 1925 : S, strings; RieT 6.1/16

Tommaso TraettaText: Giovanni Ambrogio Migliavacca. - Quelle: manuscript. - Provenienz: Fürstlich Fürstenbergische Hofbibliothek, DonaueschingenAria | a | Soprano Solo | Violini 2.e | Viola e | Basso | Del Sig. Tomaso Traetta | In Parma | 1760 | [by other hand:] Parto, ma lasci