Related papers: On the Hamilton-Waterloo problem: the case of two …

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…

The Hamilton-Waterloo Problem (HWP) in the case of $C_{m}$-factors and $C_{n}$-factors asks if $K_v$, where $v$ is odd (or $K_v-F$, where $F$ is a 1-factor and $v$ is even), can be decomposed into r copies of a 2-factor made either entirely…

The Hamilton-Waterloo Problem HWP$(v;m,n;\alpha,\beta)$ asks for a 2-factorization of the complete graph $K_v$ or $K_v-I$, the complete graph with the edges of a 1-factor removed, into $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors, where…

The Hamilton-Waterloo problem with uniform cycle sizes asks for a $2-$ factorization of the complete graph $K_v$ (for odd {\em v}) or $K_v$ minus a $1-$factor (for even {\em v}) where $r$ of the factors consist of $n-$cycles and $s$ of the…

The Hamilton-Waterloo problem asks for a 2-factorization of $K_v$ (for $v$ odd) or $K_v$ minus a $1$-factor (for $v$ even) into $C_m$-factors and $C_n$-factors. We completely solve the Hamilton-Waterloo problem in the case of $C_3$-factors…

Let $K_v^*$ denote the complete graph $K_v$ if $v$ is odd and $K_v-I$, the complete graph with the edges of a 1-factor removed, if $v$ is even. Given non-negative integers $v, M, N, \alpha, \beta$, the Hamilton-Waterloo problem asks for a…

The Directed Hamilton-Waterloo Problem asks for a directed $2$-factorization of the complete symmetric digraph $K_v^*$ where there are two non-isomorphic $2$-factors. In the uniform version of the problem, factors consist of either directed…

The uniform Hamilton-Waterloo Problem (HWP) asks for a resolvable $(C_M, C_N)$-decomposition of $K_v$ into $\alpha$ $C_M$-factors and $\beta$ $C_N$-factors. We denote a solution to the uniform Hamilton Hamilton-Waterloo problem by…

Given 2-factors $R$ and $S$ of order $n$, let $r$ and $s$ be nonnegative integers with $r+s=\lfloor \frac{n-1}{2}\rfloor$, the Hamilton-Waterloo problem asks for a 2-factorization of $K_n$ if $n$ is odd, or of $K_n-I$ if $n$ is even, in…

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…

The Hamilton-Waterloo problem is a problem of graph factorization. The Hamilton-Waterloo problem HWP$(H;m,n;\alpha,\beta)$ asks for a $2$-factorization of $H$ containing $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors. In this paper, we…

We construct new resolvable decompositions of complete multigraphs and complete equipartite multigraphs into cycles of variable lengths (and a perfect matching if the vertex degrees are odd). We develop two techniques: {\em layering}, which…

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 $n = 2^a3^b$…

Given non-negative integers $v, m, n, \alpha, \beta$, the Hamilton-Waterloo problem asks for a factorization of the complete graph $K_v$ into $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors. Clearly, $v$ odd, $n,m\geq 3$, $m\mid v$, $n\mid…

In this paper we give a complete solution to the Hamilton-Waterloo problem for the case of Hamilton cycles and C4k-factors for all positive integers k.

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…

In this paper, we almost completely solve the Hamilton-Waterloo problem with C8- factors and Cm-factors where the number of vertices is a multiple of 8m.

We prove that $K_n+I$, the complete graph of an even order with a $1$-factor duplicated, admits a decomposition into $2$-factors, each a disjoint union of cycles of length $m \geq 5$ if and only if $m \mid n$, except possibly when $m$ is…

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…

A Hamiltonian decomposition of a regular graph is a partition of its edge set into Hamiltonian cycles. The problem of finding edge-disjoint Hamiltonian cycles in a given regular graph has many applications in combinatorial optimization and…