Oberwolfach Problem Research Articles

The Oberwolfach problem with loving couples

AbstractWe generalize the well‐known Oberwolfach problem posed by Ringel in 1967. We suppose to have couples (here is an even integer) and suppose that they have to be seated for several nights at round tables in such a way that each person seats next to his partner exactly times and next to every other person exactly once. We call this problem the Oberwolfach problem with loving couples. When , the problem coincides with the so‐called spouse‐avoiding variant, which was introduced by Huang, Kotzig, and Rosa in 1979. While if either or equals the number of nights, it corresponds to the spouse‐loving variant or to the Honeymoon variant, which was recently studied by Bolohan et al. and by Lepine and Sajna, respectively. In this paper, for each possible choice of , we construct many classes of solutions to the Oberwolfach problem with loving couples. We also obtain new solutions to the Honeymoon variant.

Completing the solution of the directed Oberwolfach problem with cycles of equal length

AbstractIn 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 equal odd length. We prove that the complete symmetric digraph on vertices, denoted , admits a resolvable decomposition into directed cycles of odd length . This completely settles the directed Oberwolfach problem with tables of uniform length.

On the directed Oberwolfach problem for complete symmetric equipartite digraphs and uniform‐length cycles

AbstractWe examine the necessary and sufficient conditions for a complete symmetric equipartite digraph with parts of size to admit a resolvable decomposition into directed cycles of length . We show that the obvious necessary conditions are sufficient for in each of the following four cases: (i) is even; (ii) ; (iii) and or ; and (iv) , and if , then for a prime .

On the Oberwolfach problem for single-flip 2-factors via graceful labelings

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 Kv into copies of F. In this paper we show that OP(F) has a solution whenever F has a sufficiently large cycle which meets a given lower bound and, in addition, has a single-flip automorphism, which is an involutory automorphism acting as a reflection on exactly one of the cycles of F. Furthermore, we prove analogous results for the minimum covering version and the maximum packing version of the problem. We also show a similar result when the edges of Kv have multiplicity 2, but in this case we do not require that F be single-flip.Our approach allows us to explicitly construct solutions to the Oberwolfach Problem with well-behaved automorphisms, in contrast with some recent asymptotic results, based on probabilistic methods, which are nonconstructive and do not provide a lower bound on the order of F that guarantees the solvability of OP(F).Our constructions are based on a doubling construction which applies to graceful labelings of 2-regular graphs with a vertex removed. We show that this class of graphs is graceful as long as the length of the path-component is sufficiently large. A much better lower bound on the length of the path is given for an α-labeling of such graphs to exist.

The generalised Oberwolfach problem

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 Oberwolfach problem.

A greedy algorithm for the social golfer and the Oberwolfach problem

Inspired by the increasing popularity of Swiss-system tournaments in sports, we study the problem of predetermining the number of rounds that can be guaranteed in a Swiss-system tournament. Matches of these tournaments are usually determined in a myopic round-based way dependent on the results of previous rounds. Together with the hard constraint that no two players meet more than once during the tournament, at some point it might become infeasible to schedule a next round. For tournaments with n players and match sizes of k≥2 players, we prove that we can always guarantee ⌊nk(k−1)⌋ rounds. We show that this bound is tight. This provides a simple polynomial time constant factor approximation algorithm for the social golfer problem.We extend the results to the Oberwolfach problem. We show that a simple greedy approach guarantees at least ⌊n+46⌋ rounds for the Oberwolfach problem. This yields a polynomial time 13+ε-approximation algorithm for any fixed ε>0 for the Oberwolfach problem. Assuming that El-Zahar’s conjecture is true, we improve the bound on the number of rounds to be essentially tight.

Resolvable cycle decompositions of complete multigraphs and complete equipartite multigraphs via layering and detachment

AbstractWe 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: layering, which allows us to obtain 2‐factorizations of complete multigraphs from existing 2‐factorizations of complete graphs, and detachment, which allows us to construct resolvable cycle decompositions of complete equipartite multigraphs from existing resolvable cycle decompositions of complete multigraphs. These techniques are applied to obtain new 2‐factorizations of a specified type for both complete multigraphs and complete equipartite multigraphs, with the emphasis on new solutions to the Oberwolfach Problem and the Hamilton–Waterloo Problem. In addition, we show existence of some thickly resolvable cycle decompositions.

Resolution of the Oberwolfach problem

The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of K_{2n+1} into edge-disjoint copies of a given 2-factor. We show that this can be achieved for all large n . We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton–Waterloo problem for large n .

Solution to the outstanding case of the spouse‐loving variant of the Oberwolfach problem with uniform cycle length

AbstractLet denote the complete graph of even order with a 1‐factor duplicated. The spouse‐loving variant of the Oberwolfach Problem, denoted , asks for the existence of a 2‐factorization of in which each 2‐factor consists of cycles of length , for all , such that . If , then the problem is denoted by . In this paper, we construct a solution to when is an odd integer. This completes the proof of the conjecture posed by Bolohan et al. In addition, we find a solution to when is an odd integer.

Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem.

A complete solution to the infinite Oberwolfach problem

AbstractLet 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 the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group G. We will also consider the same problem in the more general context of graphs F that are spanning subgraphs of an infinite complete graph and we provide a solution when F is locally finite. Moreover, we characterize the infinite subgraphs L of F such that there exists a solution to OP(F) containing a solution to OP(L).

On the generalized Oberwolfach problem

The generalized Oberwolfach problem OP t (2 w + 1; N 1 , N 2 , …, N t ; α 1 , α 2 , …, α t ) asks for a factorization of K 2 w + 1 into α i C N i -factors (where a C N i -factor of K 2 w + 1 is a spanning subgraph whose components are cycles of length N i ≥ 3 ) for i = 1, 2, …, t . Necessarily, N = lcm( N 1 , N 2 , …, N t ) is a divisor of 2 w + 1 and w = Σ t i = 1 α i . For t = 1 we have the classic Oberwolfach problem. For t = 2 this is the well-studied Hamilton-Waterloo problem, whereas for t ≥ 3 very little is known. In this paper, we show, among other things, that the above necessary conditions are sufficient whenever 2 w + 1 ≥ ( t + 1) N , α i > 1 for every i ∈ {1, 2, …, t } , and gcd ( N 1 , N 2 , …, N t ) > 1 . We also provide sufficient conditions for the solvability of the generalized Oberwolfach problem over an arbitrary graph and, in particular, the complete equipartite graph.

On the honeymoon Oberwolfach problem

AbstractThe honeymoon Oberwolfach problem HOP asks the following question. Given newlywed couples at a conference and round tables of sizes , is it possible to arrange the participants at these tables for meals so that each participant sits next to their spouse at every meal and sits next to every other participant exactly once? A solution to HOP is a decomposition of , the complete graph with additional copies of a fixed 1‐factor , into 2‐factors, each consisting of disjoint ‐alternating cycles of lengths . It is also equivalent to a semi‐uniform 1‐factorization of of type ; that is, a 1‐factorization such that for all , the 2‐factor consists of disjoint cycles of lengths . In this paper, we first introduce the honeymoon Oberwolfach problem and then present several results. Most notably, we completely solve the case with uniform cycle lengths, that is, HOP. In addition, we show that HOP has a solution in each of the following cases: ; is odd and ; as well as for all . We also show that HOP has a solution whenever is odd and the Oberwolfach problem with tables of sizes has a solution.

A blow-up lemma for approximate decompositions

We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to long-standing decomposition problems. For instance, our results imply the following. Let G G be a quasi-random n n -vertex graph and suppose H 1 , … , H s H_1,\dots ,H_s are bounded degree n n -vertex graphs with ∑ i = 1 s e ( H i ) ≤ ( 1 − o ( 1 ) ) e ( G ) \sum _{i=1}^{s} e(H_i) \leq (1-o(1)) e(G) . Then H 1 , … , H s H_1,\dots ,H_s can be packed edge-disjointly into G G . The case when G G is the complete graph K n K_n implies an approximate version of the tree packing conjecture of Gyárfás and Lehel for bounded degree trees, and of the Oberwolfach problem. We provide a more general version of the above approximate decomposition result which can be applied to super-regular graphs and thus can be combined with Szemerédi’s regularity lemma. In particular our result can be viewed as an extension of the classical blow-up lemma of Komlós, Sárkőzy, and Szemerédi to the setting of approximate decompositions.

A method to construct 1‐rotational factorizations of complete graphs and solutions to the oberwolfach problem

AbstractThe concept of a 1‐rotational factorization of a complete graph under a finite group was studied in detail by Buratti and Rinaldi. They found that if admits a 1‐rotational 2‐factorization, then the involutions of are pairwise conjugate. We extend their result by showing that if a finite group admits a 1‐rotational ‐factorization with even and odd, then has at most conjugacy classes containing involutions. Also, we show that if has exactly conjugacy classes containing involutions, then the product of a central involution with an involution in one conjugacy class yields an involution in a different conjugacy class. We then demonstrate a method of constructing a 1‐rotational ‐factorization under given a 1‐rotational 2‐factorization under a finite group . This construction, given a 1‐rotational solution to the Oberwolfach problem , allows us to find a solution to when the ’s are even (), and when is an odd prime, with no restrictions on the ’s.

A bandwidth theorem for approximate decompositions

We provide a degree condition on a regular n ‐vertex graph G which ensures the existence of a near optimal packing of any family H of bounded degree n ‐vertex k ‐chromatic separable graphs into G . In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of Bottcher, Schacht and Taraz in the setting of approximate decompositions. More precisely, let δ k be the infimum over all δ ⩾ 1 / 2 ensuring an approximate K k ‐decomposition of any sufficiently large regular n ‐vertex graph G of degree at least δ n . Now suppose that G is an n ‐vertex graph which is close to r ‐regular for some r ⩾ ( δ k + o ( 1 ) ) n and suppose that H 1 , ⋯ , H t is a sequence of bounded degree n ‐vertex k ‐chromatic separable graphs with ∑ i e ( H i ) ⩽ ( 1 − o ( 1 ) ) e ( G ) . We show that there is an edge‐disjoint packing of H 1 , ⋯ , H t into G . If the H i are bipartite, then r ⩾ ( 1 / 2 + o ( 1 ) ) n is sufficient. In particular, this yields an approximate version of the tree packing conjecture in the setting of regular host graphs G of high degree. Similarly, our result implies approximate versions of the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree.

Cycle frames and the Oberwolfach problem

For an integer λ,G(λ) denotes a graph G with uniform edge-multiplicity λ. Let J be a subset of positive integers. A 2-regular subgraph of m-partite graph Km⊗In containing vertices of all but one partite set is called partial 2-factor, where ⊗ denotes wreath product and In is an independent set on n vertices. If (Km⊗In)(λ) can be partitioned into edge-disjoint partial 2-factors such that each partial 2-factor consists of cycles of lengths from J, then we say that (Km⊗In)(λ)has a(J,λ)-cycle frame. The Oberwolfach problem OP(m1α1,m2α2,…,mtαt), raised by Ringel, asks the existence of a 2-factorization of Kn (when n is odd) or Kn−I (when n is even), in which each 2-factor consists of exactly αi cycles of length mi, i=1,2,…,t. In this paper, we show that there exists a ({4,6},λ)-cycle frame of (Km⊗In)(λ) if and only if λn≡0(mod2), m≥3, (n,m)∉{(1,3),(2t+1,2s)∣t≥0,s≥2}. Further we show that there exists a ({3,4},1)-cycle frame of Km⊗In if and only if m≥3 and n≡0(mod2). As a consequence, we solve OP(3a,4b), OP(3a,6b) and OP(5a,10b) with some restrictions on a,b∈N.

Cyclic Uniform 2-Factorizations of the Complete Multipartite Graph

A generalization of the Oberwolfach problem, proposed by Liu (J Comb Des 8:42–49, 2000), asks for a uniform 2-factorization of the complete multipartite graph $$K_{m\times n}$$ . Here we focus our attention on cyclic 2-factorizations, whose 2-factors are disjoint union of cycles all of even length $$\ell $$ . In particular, we present a complete solution for the extremal cases $$\ell =4$$ and $$\ell =mn$$ .

Decomposing the complete graph and the complete graph minus a 1-factor into copies of a graph G where G is the union of two disjoint cycles

Let G of order n be the vertex-disjoint union of two cycles. It is known that there exists a G-decomposition of Kv for all v≡1(mod2n). If G is bipartite and x is a positive integer, it is also known that there exists a G-decomposition of Knx−I, where I is a 1-factor. If G is not bipartite, there exists a G-decomposition of Kn if n is odd, and of Kn−I, where I is a 1-factor, if n is even. We use novel extensions of the Bose construction for Steiner triple systems and some recent results on the Oberwolfach Problem to obtain a G-decomposition of Kv for all v≡n(mod2n) when n is odd, unless G=C4∪C5 and v=9. If G consists of two odd cycles and n≡0(mod4), we also obtain a G-decomposition of Kv−I, for all v≡0(modn), v≠4n.

Packing spanning graphs from separable families

Let $$\mathcal{G}$$ be a separable family of graphs. Then for all positive constants ϵ and Δ and for every sufficiently large integer n, every sequence G 1,..., G t ∈ $$\mathcal{G}$$ of graphs of order n and maximum degree at most Δ such that $$\left( {{G_1}} \right) + \cdots + e\left( {{G_t}} \right) \leqslant \left( {1 - \epsilon } \right)\left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right)$$ packs into K n . This improves results of Böttcher, Hladký, Piguet and Taraz when $$\mathcal{G}$$ is the class of trees and of Messuti, Rödl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of Gyárfás and Lehel (1976) for the case that all trees have maximum degree at most Δ. The proof uses the local resilience of random graphs and a special multi-stage packing procedure.