31 research outputs found

### 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

### Constructing generalized Heffter arrays via near alternating sign matrices

Let $S$ be a subset of a group $G$ (not necessarily abelian) such that
$S\,\cap -S$ is empty or contains only elements of order $2$, and let
$\mathbf{h}=(h_1,\ldots, h_m)\in \mathbb{N}^m$ and $\mathbf{k}=(k_1, \ldots,
k_n)\in \mathbb{N}^n$. A generalized Heffter array GHA$^{\lambda}_S(m, n;
\mathbf{h}, \mathbf{k})$ over $G$ is an $m\times n$ matrix $A=(a_{ij})$ such
that: the $i$-th row (resp. $j$-th column) of $A$ contains exactly $h_i$ (resp.
$k_j$) nonzero elements, and the list $\{a_{ij}, -a_{ij}\mid a_{ij}\neq 0\}$
equals $\lambda$ times the set $S\,\cup\, -S$. We speak of a zero sum (resp.
nonzero sum) GHA if each row and each column of $A$ sums to zero (resp. a
nonzero element), with respect to some ordering.
In this paper, we use near alternating sign matrices to build both zero and
nonzero sum GHAs, over cyclic groups, having the further strong property of
being simple. In particular, we construct zero sum and simple GHAs whose row
and column weights are congruent to $0$ modulo $4$. This result also provides
the first infinite family of simple (classic) Heffter arrays to be rectangular
($m\neq n$) and with less than $n$ nonzero entries in each row. Furthermore, we
build nonzero sum GHA$^{\lambda}_S(m, n; \mathbf{h}, \mathbf{k})$ over an
arbitrary group $G$ whenever $S$ contains enough noninvolutions, thus extending
previous nonconstructive results where $\pm S = G\setminus H$ for some subgroup
$H$~of~$G$.
Finally, we describe how GHAs can be used to build orthogonal decompositions
and biembeddings of Cayley graphs (over groups not necessarily abelian) onto
orientable surfaces.Comment: 29 pages, 1 figur

### Vertex-regular $1$-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