A Generalization of the Hamilton-Waterloo Problem on Complete Equipartite Graphs

The Hamilton-Waterloo problem asks for which $s$ and $r$ the complete graph $K_n$ can be decomposed into $s$ copies of a given 2-factor $F_1$ and $r$ copies of a given 2-factor $F_2$ (and one copy of a 1-factor if $n$ is even). In this paper we generalize the problem to complete equipartite graphs $K_{(n:m)}$ and show that $K_{(xyzw:m)}$ can be decomposed into $s$ copies of a 2-factor consisting of cycles of length $xzm$; and $r$ copies of a 2-factor consisting of cycles of length $yzm$, whenever $m$ is odd, $s,r\neq 1$, $\gcd(x,z)=\gcd(y,z)=1$ and $xyz\neq 0 \pmod 4$. We also give some more general constructions where the cycles in a given two factor may have different lengths. We use these constructions to find solutions to the Hamilton-Waterloo problem for complete graphs

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 Hamilton-Waterloo problem: the case of two cycles sizes of different parity

The Hamilton-Waterloo problem asks for a decomposition of the complete graph of order v into r copies of a 2-factor F1 and s copies of a 2-factor F2 such that r+s = v−1 2 . If F1 consists of m-cycles and F2 consists of n cycles, we say that a solution to (m, n)- HWP(v; r, s) exists. The goal is to find a decomposition for every possible pair (r, s). In this paper, we show that for odd x and y, there is a solution to (2kx, y)-HWP(vm; r, s) if gcd(x, y) ≥ 3, m ≥ 3, and both x and y divide v, except possibly when 1 ∈ {r, s}

Two Problems of Gerhard Ringel

Gerhard Ringel was an Austrian Mathematician, and is regarded as one of the most influential graph theorists of the twentieth century. This work deals with two problems that arose from Ringel\u27s research: the Hamilton-Waterloo Problem, and the problem of R-Sequences. The Hamilton-Waterloo Problem (HWP) in the case of Cm-factors and Cn-factors asks whether Kv, where v is odd (or Kv-F, where F is a 1-factor and v is even), can be decomposed into r copies of a 2-factor made entirely of m-cycles and s copies of a 2-factor made entirely of n-cycles. Chapter 1 gives some general constructions for such decompositions and apply them to the case where m=3 and n=3x. This problem is settle for odd v, except for a finite number of x values. When v is even, significant progress is made on the problem, although open cases are left. In particular, the difficult case of v even and s=1 is left open for many situations. Chapter 2 generalizes the Hamilton-Waterloo Problem to complete equipartite graphs K(n:m) and shows that K(xyzw:m) can be decomposed into s copies of a 2-factor consisting of cycles of length xzm and r copies of a 2-factor consisting of cycles of length yzm, whenever m is odd, s,r≠1, gcd(x,z)=gcd(y,z)=1 and xyz≠0 (mod 4). Some more general constructions are given for the case when the cycles in a given two factor may have different lengths. These constructions are used to find solutions to the Hamilton-Waterloo problem for complete graphs. Chapter 3 completes the proof of the Friedlander, Gordon and Miller Conjecture that every finite abelian group whose Sylow 2-subgroup either is trivial or both non-trivial and non-cyclic is R-sequenceable. This settles a question of Ringel for abelian groups

Cyclic cycle systems of the complete multipartite graph

In this paper, we study the existence problem for cyclic $\ell$-cycle decompositions of the graph $K_m[n]$, the complete multipartite graph with $m$ parts of size $n$, and give necessary and sufficient conditions for their existence in the case that $2\ell \mid (m-1)n$

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

A generalization of Heffter arrays

In this paper we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let $v=2nk+t$ be a positive integer, where $t$ divides $2nk$, and let $J$ be the subgroup of $\mathbb{Z}_v$ of order $t$. A $H_t(m,n; s,k)$ Heffter array over $\mathbb{Z}_v$ relative to $J$ is an $m\times n$ partially filled array with elements in $\mathbb{Z}_v$ such that: (a) each row contains $s$ filled cells and each column contains $k$ filled cells; (b) for every $x\in \mathbb{Z}_v\setminus J$, either $x$ or $-x$ appears in the array; (c) the elements in every row and column sum to $0$. Here we study the existence of square integer (i.e. with entries chosen in $\pm\left\{1,\dots,\left\lfloor \frac{2nk+t}{2}\right\rfloor \right\}$ and where the sums are zero in $\mathbb{Z}$) relative Heffter arrays for $t=k$, denoted by $H_k(n;k)$. In particular, we prove that for $3\leq k\leq n$, with $k\neq 5$, there exists an integer $H_k(n;k)$ if and only if one of the following holds: (a) $k$ is odd and $n\equiv 0,3\pmod 4$; (b) $k\equiv 2\pmod 4$ and $n$ is even; (c) $k\equiv 0\pmod 4$. Also, we show how these arrays give rise to cyclic cycle decompositions of the complete multipartite graph