Related papers: $m$-Cycle Packings of $(\lambda+\mu)K_{v+u}-\lambd…

Following standard terminology, $\lambda K_v$ is a multigraph on $v$ vertices such that $\lambda$ edges join each pair of vertices. Let $(\lambda+\mu)K_{v+u}-\lambda K_v$ be the graph $(V\cup U,E)$ with $|V|=v$, $|U|=u$, and…

In a graph $G$, let $\mu_G(xy)$ denote the number of edges between $x$ and $y$ in $G$. Let $\lambda K_{v,u}$ be the graph $(V\cup U,E)$ with $|V|=v$, $|U|=u$, and \[ \mu_G(xy)=\begin{cases} \lambda &\mbox{if $x\in U$ and $y\in V$ or if…

Denote by $\lambda K_v$ the complete graph of order $v$ with multiplicity $\lambda$. Let $\lambda K_v-\lambda K_w-\lambda K_u$ be the graph obtained from $\lambda K_v$ by the removal of the edges of two vertex disjoint complete…

We prove that the complete graph with a hole $K_{u+w}-K_u$ can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, each cycle has length at most $\min(u,w)$, and the longest…

A cycle of length $t$ in a hypergraph is an alternating sequence $v_1,e_1,v_2\dots,v_t,e_t$ of distinct vertices $v_i$ and distinct edges $e_i$ so that $\{v_i,v_{i+1}\}\subseteq e_i$ (with $v_{t+1}:=v_1$). Let $\lambda K_n^h$ be the…

A graph with $v$ vertices is $(r)$-pancyclic if it contains precisely $r$ cycles of every length from 3 to $v$. A bipartite graph with even number of vertices $v$ is said to be $(r)$-bipancyclic if it contains precisely $r$ cycles of each…

A $k$-uniform tight cycle is a $k$-graph with a cyclic order of its vertices such that every $k$ consecutive vertices from an edge. We show that for $k\geq 3$, every red-blue edge-coloured complete $k$-graph on $n$ vertices contains $k$…

A hamiltonian cycle system (HCS, for short) of a graph $\Gamma$ is a partition of the edges of $\Gamma$ into hamiltonian cycles. A HCS is cyclic when it is invariant under a cyclic permutation of all the vertices of $\Gamma$; the existence…

Let $K_v$ denote the complete graph of order $v$ and $K_v - I$ denote $K_v$ minus a 1-factor. In this article we investigate uniformly resolvable decompositions of $K_v$ and $K_v-I$ into $r$ classes containing only copies of $3$-stars and…

For each odd $m \geq 3$ we completely solve the problem of when an $m$-cycle system of order $u$ can be embedded in an $m$-cycle system of order $v$, barring a finite number of possible exceptions. In cases where $u$ is large compared to…

Using the technique of amalgamation-detachment, we show that the complete equipartite multigraph $\lambda K_{n\times m}$ can be decomposed into cycles of lengths $c_1m,\dots,c_km$ (plus a 1-factor if the degree is odd) whenever there exists…

In this paper, we provide necessary and sufficient conditions for the existence of a cyclic $m$-cycle system of $K_n-I$ when $m$ and $n$ are even and $m \mid n$.

We show that the complete graph on $n$ vertices can be decomposed into $t$ cycles of specified lengths $m_1,\ldots,m_t$ if and only if $n$ is odd, $3\leq m_i\leq n$ for $i=1,\ldots,t$, and $m_1+\cdots+m_t=\binom n2$. We also show that the…

In this paper, we prove that almost resolvable $k$-cycle systems (briefly $k$-ARCS) of $(K_u \times K_g)(\lambda)$ exists for all $k \equiv 0(mod \ 4) $ with few possible exceptions, where $\times$ represents tensor product of graphs.

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…

We consider edge decompositions of $K_v^{(3)}-I$, the complete 3-uniform hypergraph of order $v$ minus a 1-factor (parallel class, packing of $v/3$ disjoint edges). We prove that a decomposition into tight 6-cycles exists if and only if…

Bryant, Horsley, Maenhaut and Smith recently gave necessary and sufficient conditions for when the complete multigraph can be decomposed into cycles of specified lengths $m_1,m_2,\ldots,m_\tau$. In this paper we characterise exactly when…

A $k$-cycle with a pendant edge attached to each vertex is called a $k$-sun. The existence problem for $k$-sun decompositions of $K_v$, with $k$ odd, has been solved only when $k=3$ or $5$. By adapting a method used by Hoffmann, Lindner and…

In a graph, $k$ cycles are {\em admissible} if their lengths form an arithmetic progression with common difference one or two. Let $G$ be a 2-connected graph with minimum degree at least $k\geqslant 4$. We prove that \begin{itemize} \item…

The enhanced hypercube $Q_{n,k}$ is a variant of the hypercube $Q_n$. We investigate all the lengths of cycles that an edge of the enhanced hypercube lies on. It is proved that every edge of $Q_{n,k}$ lies on a cycle of every even length…