相关论文: A complete solution to the directed Oberwolfach pr…

Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem, $OP(F)$, asks for a $2$-factorization of the complete graph on $v$ vertices in which each $2$-factor is isomorphic to $F$. In this paper, we give a complete solution to…

In this paper, we give a solution to the last outstanding case of the directed Oberwolfach problem with tables of uniform length. Namely, we address the two-table case with tables of odd length. We prove that the complete symmetric digraph…

The Directed Oberwolfach Problem can be considered as the directed version of the well-known Oberwolfach Problem, first mentioned by Ringel at a conference in Oberwolfach, Germany in 1967. In this paper, we describe some new partial results…

Various results on factorisations of complete graphs into circulant graphs and on 2-factorisations of these circulant graphs are proved. As a consequence, a number of new results on the Oberwolfach Problem are obtained. For example, a…

We address the last outstanding case of the directed Oberwolfach problem with two tables of different lengths. Specifically, we show that the complete symmetric directed graph $K^*_n$ admits a decomposition into spanning subdigraphs…

We prove that any quasirandom dense large graph in which all degrees are equal and even can be decomposed into any given collection of two-factors (2-regular spanning subgraphs). A special case of this result gives a new solution to the…

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 generalization of the Oberwolfach Problem, proposed by J. Liu in 2000, asks for a uniform $2$-factorization of the complete multipartite graph $K_{m\times n}$. Here we focus our attention on $2$-factorizations regular under the cyclic…

In this paper, we give the following result: If $D$ is a digraph of order $n$, and if $d_{D}^{+}(u) + d_{D}^{-}(v) \ge n$ for every two distinct vertices $u$ and $v$ with $(u, v) \notin A(D)$, then $D$ has a directed $2$-factor with exactly…

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…

We show that the complete symmetric digraph $K_{2m}^\ast$ admits a resolvable decomposition into directed cycles of length $m$ for all odd $m$, $5 \le m \le 49$. Consequently, $K_{n}^\ast$ admits a resolvable decomposition into directed…

The directed Oberwolfach problem OP$^\ast(m_1,\ldots,m_k)$ asks whether the complete symmetric digraph $K_n^\ast$, assuming $n=m_1+\ldots +m_k$, admits a decomposition into spanning subdigraphs, each a disjoint union of $k$ directed cycles…

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…

Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem $OP(F)$, posed in 1967 and still open, asks for a decomposition of $K_v$ into copies of $F$. In this paper we show that $OP(F)$ has a solution whenever $F$ has a…

For a graph (undirected, directed, or mixed), a cycle-factor is a collection of vertex-disjoint cycles covering the entire vertex set. Cycle-factors subject to parity constraints arise naturally in the study of structural graph theory and…

We examine the necessary and sufficient conditions for a complete symmetric equipartite digraph $K_{n[m]}^\ast$ with $n$ parts of size $m$ to admit a resolvable decomposition into directed cycles of length $t$. We show that the obvious…

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…

Let D be a directed graph with vertex set V and order n. An anti-directed hamiltonian cycle H in D is a hamiltonian cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. An anti-directed…

The concept of a $1$-rotational factorization of a complete graph under a finite group $G$ was studied in detail by Buratti and Rinaldi. They found that if $G$ admits a $1$-rotational $2$-factorization, then the involutions of $G$ are…

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…