Related papers: On the Hamilton-Waterloo Problem with cycle length…

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…

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…

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…

The Hamilton-Waterloo problem asks for a decomposition of the complete graph into $r$ copies of a 2-factor $F_{1}$ and $s$ copies of a 2-factor $F_{2}$ such that $r+s=\left\lfloor\frac{v-1}{2}\right\rfloor$. If $F_{1}$ consists of…

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 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 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 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…

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…

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.

In this paper, factorizations of the complete symmetric digraph $K_v^*$ into uniform factors consisting of directed even cycle factors are studied as a generalization of the undirected Hamilton-Waterloo Problem. It is shown, with a few…

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.

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…

For all integers $k$ with $k\geq 2$, if $G$ is a balanced $k$-partite graph on $n\geq 3$ vertices with minimum degree at least \[…

We consider the random graph $G_{n, {\bf d}}$ chosen uniformly at random from the set of all graphs with a given sparse degree sequence ${\bf d}$. We assume ${\bf d}$ has minimum degree at least 4, at most a power law tail, and place one…

Let $k\geq 2$. We show that, for a sufficiently small $\varepsilon>0$, any sufficiently large $n$-vertex Hamiltonian graph of minimum degree at least $n^{1-\varepsilon}$ contains a $2$-factor consisting of exactly $k$ cycles. This is the…

Let $k$, $m$ and $r$ be three integers such that $2\leq k\leq m\leq r$. Let $G$ be a $2r$-regular, $2m$-edge-connected graph of odd order. We obtain some sufficient conditions, which guarantee $G-v$ contains a $k$-factor for all $v\in…

We study $M$-alternating Hamilton paths and $M$-alternating Hamilton cycles in a simple connected graph $G$ on $\nu$ vertices with a perfect matching $M$. Let $G$ be a bipartite graph, we prove that if for any two vertices $x$ and $y$ in…