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 existence of a cyclic near-Rrsolvable (6n+4)-cycle system of 2K12n+9

In this article, we prove the existence of a simple cyclic near-resolvable - cycle system of for by the method of constructing its starter. Then, some new properties and results related to this construction are formulated

Ramsey-nice families of graphs

For a finite family $\mathcal{F}$ of fixed graphs let $R_k(\mathcal{F})$ be the smallest integer $n$ for which every $k$-coloring of the edges of the complete graph $K_n$ yields a monochromatic copy of some $F\in\mathcal{F}$. We say that $\mathcal{F}$ is $k$-nice if for every graph $G$ with $\chi(G)=R_k(\mathcal{F})$ and for every $k$-coloring of $E(G)$ there exists a monochromatic copy of some $F\in\mathcal{F}$. It is easy to see that if $\mathcal{F}$ contains no forest, then it is not $k$-nice for any $k$. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs $\mathcal{F}$ that contains at least one forest, and for all $k\geq k_0(\mathcal{F})$ (or at least for infinitely many values of $k$), $\mathcal{F}$ is $k$-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family $\mathcal{F}$ containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni, Charbit and Howard regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure

The directed Oberwolfach problem with variable cycle lengths: a recursive construction

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 of lengths $m_1,\ldots,m_k$. We hereby describe a method for constructing a solution to OP$^\ast(m_1,\ldots,m_k)$ given a solution to OP$^\ast(m_1,\ldots,m_\ell)$, for some $\ell<k$, if certain conditions on $m_1,\ldots,m_k$ are satisfied. This approach enables us to extend a solution for OP$^\ast(m_1,\ldots,m_\ell)$ into a solution for OP$^\ast(m_1,\ldots,m_\ell,t)$, as well as into a solution for OP$^\ast(m_1,\ldots,m_\ell,2^{\langle t \rangle})$, where $2^{\langle t \rangle}$ denotes $t$ copies of 2, provided $t$ is sufficiently large. In particular, our recursive construction allows us to effectively address the two-table directed Oberwolfach problem. We show that OP$^\ast(m_1,m_2)$ has a solution for all $2 \le m_1\le m_2$, with a definite exception of $m_1=m_2=3$ and a possible exception in the case that $m_1 \in \{ 4,6 \}$, $m_2$ is even, and $m_1+m_2 \ge 14$. It has been shown previously that OP$^\ast(m_1,m_2)$ has a solution if $m_1+m_2$ is odd, and that OP$^\ast(m,m)$ has a solution if and only if $m \ne 3$. In addition to solving many other cases of OP$^\ast$, we show that when $2 \le m_1+\ldots +m_k \le 13$, OP$^\ast(m_1,\ldots,m_k)$ has a solution if and only if $(m_1,\ldots,m_k) \not\in \{ (4),(6),(3,3) \}$