The honeymoon Oberwolfach problem HOP\((2m_1,2m_2,\ldots,2m_t)\) asks the following question. Given \(n=m_1+m_2+\ldots +m_t\) newlywed couples at a conference and \(t\) round tables of sizes \(2m_1,2m_2,\ldots,2m_t\), is it possible to arrange the \(2n\) participants at these tables for \(2n-2\) 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\((2m_1,2m_2,\ldots,2m_t)\) is a decomposition of \(K_{2n}+(2n-3)I\), the complete graph \(K_{2n}\) with \(2n-3\) additional copies of a fixed 1-factor \(I\), into 2-factors, each consisting of disjoint \(I\)-alternating cycles of lengths \(2m_1,2m_2,\ldots,2m_t\). The honeymoon Oberwolfach problem was introduced in a 2019 paper by Lepine and Šajna. The authors conjectured that HOP\((2m_1,2m_2,\ldots,\) \(2m_t)\) has a solution whenever the obvious necessary conditions are satisfied, and proved the conjecture for several large cases, including the uniform cycle length case \(m_1=\ldots=m_t\), and the small cases with \(n \le 9\). In the present paper, we extend the latter result to all cases with \(n \le 20\) using a computer-assisted search.
The well-known Oberwolfach problem, denoted OP\((m_1,\ldots,m_t)\), asks whether \(n=m_1+\ldots+m_t\) participants can be seated at \(t\) tables of sizes \(m_1,\ldots,m_t\) for several nights in a row so that each participant gets to sit next to every other participant exactly once. Thus, we are asking whether \(K_n\), the complete graph on \(n\) vertices, admits a 2-factorization such that each 2-factor is a disjoint union of \(t\) cycles of lengths \(m_1,\ldots,m_t\). The problem has been solved in many special cases — see [1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14] — but is in general still open.
In the 2019 paper [11], Lepine and the second author introduced a new variant of the Oberwolfach problem, called the honeymoon Oberwolfach problem. This problem, denoted HOP\((m_1,\ldots,m_t)\), can be described as follows. We have \(n=\frac{1}{2}(m_1+\ldots+m_t)\) newlywed couples attending a conference and \(t\) tables of sizes \(m_1,\ldots,m_t\) (where each \(m_i \ge 3\)). Is it possible to arrange the participants at these \(t\) round tables on \(2n-2\) consecutive nights so that each couple sit together every night, and every participant sits next to every other participant exactly once?
In graph-theoretic terms we are asking whether \(K_{2n}+(2n-3)I\), the multigraph obtained from the complete graph \(K_{2n}\) by adjoining \(2n-3\) additional copies of a chosen 1-factor \(I\), admits a decomposition into 2-factors, each a vertex-disjoint union of cycles of lengths \(m_1,\ldots,m_t\), so that in each of these cycles, every other edge is a copy of an edge of \(I\). A solution to HOP\((m_1,m_2,\ldots,m_t)\) is equivalent to a semi-uniform 1-factorization of \(K_{2n}\) of type \((m_1,m_2,\ldots,m_t)\); that is, a 1-factorization \(\{ F_1,F_2,\ldots,F_{2n-1} \}\) such that for all \(i \bullet 1\), the 2-factor \(F_1 \cup F_i\) consists of disjoint cycles of lengths \(m_1,m_2,\ldots,m_t\).
If \(m_1=\ldots=m_t=m\) and \(tm=2n\), then the symbol HOP\((m_1,\ldots,m_t)\) is abbreviated as HOP\((2n;m)\). Note that if HOP\((m_1,\ldots,m_t)\) has a solution, then the \(m_i\) are all even and at least 4; these are the obvious necessary conditions.
In [11], the authors proposed the following conjecture.
Conjecture 1.1. [11] The obvious necessary conditions for HOP\((m_1,\ldots,m_t)\) to have a solution are also sufficient.
They also proved the conjecture in the following cases.
Theorem 1.2. [11] Let \(m\) and \(n\) be positive integers, \(2 \le m \le n\). Then HOP\((2n;2m)\) has a solution if and only if \(n \equiv 0 \pmod{m}\).
Theorem 1.3. [11] Let \(2 \le m_1 \le \ldots \le m_t\) be integers, and \(n=m_1+\ldots +m_t\). Then HOP\((2m_1,\ldots,2m_t)\) has a solution in each of the following cases.
(a) \(m_i \equiv 0 \pmod{4}\) for all \(i\).
(b) \(n\) is odd and OP\((m_1,\ldots,m_t)\) has a solution.
(c) \(n\) is odd and \(t=2\).
(d) \(n\) is odd, \(n < 40\), and \(m_1 \ge 3\).
(e) \(n \le 9\).
We remark that case (d) of Theorem 1.3 is extended to \(n \le 100\) by the results of [13, 12]. In this paper, we extend case (e) of Theorem 1.3, thus proving the following result.
Theorem 1.4. Let \(m_1, \ldots, m_t\) be integers with \(m_i \ge 2\) for all \(i\), and \(n=m_1+\ldots +m_t\) such that \(n \le 20\). Then HOP\((2m_1,\ldots,2m_t)\) has a solution.
To prove Theorem 1.4, we use the approach described in [11], combined with a computer-assisted search. In Sections 2 and 3, we present the relevant terminology and tools from [11], and in Section 4, we give the framework of the proof of Theorem 1.4, referring to the longer version of the paper [10] for the long list of computational results supporting the proof.
In this paper, graphs may contain parallel edges, but not loops. As usual, \(K_n\) and \(\lambda K_n\) denote the complete graph and the \(\lambda\)-fold complete graph, respectively, on \(n\) vertices. For \(m \ge 2\), the symbol \(C_m\) denotes the cycle of length \(m\), or \(m\)-cycle.
Let \(G\) be a graph, and let \(H_1,\ldots,H_t\) be subgraphs of \(G\). The collection \(\{ H_1,\ldots,H_t\}\) is called a decomposition of \(G\) if \(\{ E(H_1),\ldots,E(H_t)\}\) is a partition of \(E(G)\).
An \(r\)-factor in a graph \(G\) is an \(r\)-regular spanning subgraph of \(G\), and an \(r\)-factorization of \(G\) is a decomposition of \(G\) into \(r\)-factors. A 2-factor of \(G\) consisting of disjoint cycles of lengths \(m_1,\ldots, m_t\), respectively, is called a \((C_{m_1},\ldots,C_{m_t})\)-factor of \(G\), and a decomposition into \((C_{m_1},\ldots,C_{m_t})\)-factors is called a \((C_{m_1},\ldots,C_{m_t})\)-factorization of \(G\).
For a positive integer \(n\) and \(S \subseteq \mathbb{Z}_n^\ast\) such that \(S=-S\), we define a circulant \({\rm Circ}(n;S)\) as the graph with vertex set \(\{ x_i: i \in \mathbb{Z}_n\}\) and edge set \(\{ x_i x_{i+d}: i \in \mathbb{Z}_n, d \in S\}\). An edge of the form \(x_i x_{i+d}\) is said to be of difference \(d\). Note that an edge of difference \(d\) is also of difference \(n-d\), so we may assume that each difference is in \(\{ 1,2,\ldots, \lfloor \frac{n}{2} \rfloor \}\).
In this paper, the complete graph \(K_n\) will be viewed as the join of the circulant \({\rm Circ}(n-1;\mathbb{Z}_{n-1}^\ast)\) and the complete graph \(K_1\) with vertex \(x_{\infty}\). Thus, \(V(K_n)=\{ x_i: i \in \mathbb{Z}_{n-1}\} \cup \{ x_{\infty} \}\) and \(E(K_n)=\{ x_i x_j: i,j \in \mathbb{Z}_{n-1}, i \bullet j \} \cup \{ x_i x_{\infty}: i \in \mathbb{Z}_{n-1} \}\). If this is the case, then an edge of the form \(x_i x_{\infty}\) will be called of difference infinity.
Let \(I\) be a chosen 1-factor in the graph \(K_{2n}\). An edge of \(K_{2n}\) is said to be an \(I\)-edge if it belongs to \(E(I)\), and a non-\(I\)-edge otherwise. The symbol \(K_{2n}+\lambda I\) denotes the graph \(K_{2n}\) with \(\lambda\) additional copies of each \(I\)-edge, for a total of \(\lambda+1\) copies of each \(I\)-edge. (Note that these additional copies of \(I\)-edges of \(K_{2n}\) are then also considered to be \(I\)-edges of \(K_{2n}+\lambda I\).) A cycle \(C\) of \(K_{2n}+\lambda I\), necessarily of even length, is said to be \(I\)-alternating if the \(I\)-edges and non-\(I\)-edges along \(C\) alternate. A 2-factor (or 2-factorization) of \(K_{2n}+\lambda I\) is said to be \(I\)-alternating if each of its cycles is \(I\)-alternating.
Thus, a solution to the honeymoon Oberwolfach problem HOP\((m_1,\ldots,m_t)\) is an \(I\)-alternating \((C_{m_1},\ldots,C_{m_t})\)-factorization of \(K_{2n}+(2n-3)I\) for \(2n=m_1+m_2+\ldots +m_t\).
As in [11], we use the symbol \(4K_n^\bullet\) to denote the 4-fold complete graph with \(n\) vertices whose edges are coloured pink, blue, and black, and black edges are oriented so that each 4-set of parallel edges contains one pink edge, one blue edge, and two opposite black arcs.
Definition 3.1. [11] A 2-factorization \({\cal D}\) of \(4K_n^\bullet\) is said to be HOP if each cycle of \({\cal D}\) satisfies the following condition:
(C) any two adjacent (that is, consecutive) edges satisfy one of the following:
\(\bullet\) one is blue and the other pink; or
\(\bullet\) both are black and directed in the same way;
\(\bullet\) one is blue and the other black, directed towards the blue edge; or
\(\bullet\) one is pink and the other black, directed away from the pink edge.
Theorem 3.2. [11] Let \(m_1,\ldots,m_t\) be integers greater than 1, and let \(n=m_1+\ldots+m_t\). Then HOP\((2m_1,\ldots,2m_t)\) has a solution if and only if \(4K_n^\bullet\) admits an HOP \((C_{m_1},\ldots,C_{m_t})\)-factorization.
In the next proposition, the symbol \(2K_n^{ Circ}\) denotes the multigraph \(2K_n\) whose edges are coloured pink and black so that each 2-set of parallel edges contains one pink edge and one black edge.
Proposition 3.3. [11] Assume \(n\) is even, and let the vertex set of \(2K_n^{Circ}\) be \(\{ x_i: i \in \mathbb{Z}_{n-1}\} \cup \{ x_{\infty} \}\). Let \(\rho\) be the permutation \(\rho=(x_{\infty})(x_0 \; x_1 \; x_2 \; \ldots \; x_{n-2})\), and let \(\rho_{Circ}\) denote the permutation on the edge set of \(2K_n^{Circ}\) that is induced by \(\rho\) and that preserves the colour of the edges.
Suppose \(2K_n^{Circ}\) admits a \((C_{m_1},\ldots,C_{m_t})\)-factor \(F\) such that
each cycle in \(F\) of length at least 3 contains an even number of pink edges, and
\(F\) contains exactly one edge from each of the orbits of \(\langle \rho_{Circ} \rangle\).
Then \(4K_n^{\bullet}\) admits an HOP \((C_{m_1},\ldots,C_{m_t})\)-factorization.
Proposition 3.4. [11] Assume \(n\) is even, and let the vertex set of \(4K_n^{\bullet}\) be \(\{ x_i: i \in \mathbb{Z}_{n-1}\} \cup \{ x_{\infty} \}\). Let \(\rho\) be the permutation \(\rho=(x_{\infty})(x_0 \; x_1 \; x_2 \; \ldots \; x_{n-2})\), and let \(\rho_{\bullet}\) denote the permutation on the edge set of \(4K_n^{\bullet}\) that is induced by \(\rho\) and that preserves the colour (and orientation) of the edges.
Suppose \(4K_n^{\bullet}\) admits edge-disjoint \((C_{m_1},\ldots,C_{m_t})\)-factors \(F_1\) and \(F_2\) such that
each cycle in \(F_1\) and \(F_2\) satisfies Condition (C) in Definition 3.1, and
\(F_1\) and \(F_2\) jointly contain exactly one edge from each of the orbits of \(\langle \rho_{\bullet} \rangle\).
Then \(\mathcal{D}=\{ \rho_{\bullet}^i(F_1), \rho_{\bullet}^i(F_2): i \in \mathbb{Z}_{n-1} \}\) is an HOP \((C_{m_1},\ldots,C_{m_t})\)-factorization of \(4K_n^{\bullet}\).
Proposition 3.5. [11] Assume \(n\) is odd, and let the vertex set of \(4K_n^{\bullet}\) be \(\{ x_i: i \in \mathbb{Z}_{n-1} \} \cup \{ x_{\infty} \}\). Let \(\rho\) be the permutation \(\rho=(x_{\infty})(x_0 \; x_1 \; x_2 \; \ldots \; x_{n-2})\), and let \(\rho_{\bullet}\) denote the permutation on the edge set of \(4K_n^{\bullet}\) that is induced by \(\rho\) and that preserves the colour (and orientation) of the edges.
Suppose \(4K_n^{\bullet}\) admits pairwise edge-disjoint \((C_{m_1},\ldots,C_{m_t})\)-factors \(F_1\), \(F_2\), and \(F_3\) such that
each cycle in \(F_1\), \(F_2\), and \(F_3\) satisfies Condition (C) in Definition 3.1;
each orbit of \(\langle \rho_{\bullet} \rangle\) has edges either in \(F_1 \cup F_2\) or in \(F_3\);
if \(e \in E(F_1 \cup F_2)\), then \(\rho_{\bullet}^{\frac{n-1}{2}}(e)\in E(F_1 \cup F_2)\); and
\(F_1 \cup F_2\) contains a pink and a blue edge of difference \(\frac{n-1}{2}\).
Then \(\mathcal{D}=\{ \rho_{\bullet}^i(F_1), \rho_{\bullet}^i(F_2): i=0,1,\ldots,\frac{n-3}{2} \} \cup \{ \rho_{\bullet}^i(F_3): i \in \mathbb{Z}_{n-1} \}\) is an HOP \((C_{m_1},\ldots,C_{m_t})\)-factorization of \(4K_n^{\bullet}\).
The required 2-factors \(F\) from Proposition 3.3, \(F_1\) and \(F_2\) from Proposition 3.4, and \(F_1\), \(F_2\), and \(F_3\) from Proposition 3.5 will are called the starter 2-factors (or starters) of the resulting 2-factorizations. Thus, we are referring to Propositions 3.3, 3.4, and 3.5 as the one-starter, two-starter, and three-starter approach, respectively.
By the type of a \((C_{m_1},\ldots,C_{m_t})\)-factor we mean the multiset \([m_1,\ldots,m_t]\).
Proof. By Theorem 1.3(e), it suffices to consider \(10 \le n \le 20\). For each value of \(n\), we list all possible 2-factor types \([m_1,\ldots,m_t]\), and refer to the result that guarantees existence of a solution to HOP\((2m_1,\ldots,2m_t)\). In cases where we are referring to Proposition 3.3, 3.4, or 3.5, the computational results (that is, appropriate starter 2-factors) are given in the appendices of the extended version of this paper, namely [10].
Case \(n=10\): see [10, Appendix B] for the computational results.
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([2, 2, 2, 2, 2]\) | Theorem 1.2 |
| \([4, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([3, 3, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 2, 2]\) | Proposition 3.4 (two starters) |
| \([5, 3, 2]\) | Proposition 3.3 (one starter) |
| \([4, 4, 2]\) | Proposition 3.3 (one starter) |
| \([4, 3, 3]\) | Proposition 3.4 (two starters) |
| \([8,2]\) | Proposition 3.3 (one starter) |
| \([7,3]\) | Proposition 3.4 (two starters) |
| \([6,4]\) | Proposition 3.4 (two starters) |
| \([5,5]\) | Theorem 1.2 |
| \([10]\) | Theorem 1.2 |
Case \(n=11\): see [10, Appendix C] for the computational results.
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([3, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([4, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([3, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| \([7, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 3, 2]\) | Proposition 3.5 (three starters) |
| \([5, 4, 2]\) | Proposition 3.5 (three starters) |
| \([5, 3, 3]\) | Theorem 1.2 |
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([4, 4, 3]\) | Theorem 1.2 |
| \([9,2]\) | Theorem 1.2 |
| \([8,3]\) | Theorem 1.2 |
| \([7,4]\) | Theorem 1.2 |
| \([6,5]\) | Theorem 1.2 |
| \([11]\) | Theorem 1.2 |
Case \(n=12\): see [10, Appendix D] for the computational results.
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([2, 2, 2, 2, 2, 2]\) | Theorem 1.2 |
| \([4, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([3, 3, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([5, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([4, 4, 2, 2]\) | Proposition 3.3 (one starter) |
| \([4, 3, 3, 2]\) | Proposition 3.4 (two starters) |
| \([3, 3, 3, 3]\) | Theorem 1.2 |
| \([8, 2, 2]\) | Proposition 3.3 (one starter) |
| \([7, 3, 2]\) | Proposition 3.4 (two starters) |
| \([6, 4, 2]\) | Proposition 3.4 (two starters) |
| \([5, 5, 2]\) | Proposition 3.4 (two starters) |
| \([6, 3, 3]\) | Proposition 3.3 (one starter) |
| \([5, 4, 3]\) | Proposition 3.3 (one starter) |
| \([4, 4, 4]\) | Theorem 1.2 |
| \([10,2]\) | Proposition 3.4 (two starters) |
| \([9,3]\) | Proposition 3.3 (one starter) |
| \([8,4]\) | Theorem 1.2 |
| \([7,5]\) | Proposition 3.3 (one starter) |
| \([6,6]\) | Theorem 1.2 |
| \([12]\) | Theorem 1.2 |
Case \(n=13\): see [10, Appendix E] for the computational results.
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([3, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([4, 3, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([3, 3, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([7, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 4, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| \([4, 4, 3, 2]\) | Proposition 3.5 (three starters) |
| \([4, 3, 3, 3]\) | Theorem 1.2 |
| \([9, 2, 2]\) | Proposition 3.5 (three starters) |
| \([8, 3, 2]\) | Proposition 3.5 (three starters) |
| \([7, 4, 2]\) | Proposition 3.5 (three starters) |
| \([6, 5, 2]\) | Proposition 3.5 (three starters) |
| \([7, 3, 3]\) | Theorem 1.2 |
| \([6, 4, 3]\) | Theorem 1.2 |
| \([5, 5, 3]\) | Theorem 1.2 |
| \([5, 4, 4]\) | Theorem 1.2 |
| \([11,2]\) | Theorem 1.2 |
| \([10,3]\) | Theorem 1.2 |
| \([9,4]\) | Theorem 1.2 |
| \([8,5]\) | Theorem 1.2 |
| \([7,6]\) | Theorem 1.2 |
| \([13]\) | Theorem 1.2 |
Case \(n=14\): see [10, Appendix F] for the computational results.
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([2, 2, 2, 2, 2, 2, 2]\) | Theorem 1.2 |
| \([4, 2, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([3, 3, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([5, 3, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([4, 4, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([4, 3, 3, 2, 2]\) | Proposition 3.4 (two starters) |
| \([3, 3, 3, 3, 2]\) | Proposition 3.3 (one starter) |
| \([8, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([7, 3, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 4, 2, 2]\) | Proposition 3.4 (two starters) |
| \([5, 5, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 3, 3, 2]\) | Proposition 3.3 (one starter) |
| \([5, 4, 3, 2]\) | Proposition 3.3 (one starter) |
| \([4, 4, 4, 2]\) | Proposition 3.3 (one starter) |
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([5, 3, 3, 3]\) | Proposition 3.4 (two starters) |
| \([4, 4, 3, 3]\) | Proposition 3.4 (two starters) |
| \([10, 2, 2]\) | Proposition 3.4 (two starters) |
| \([9, 3, 2]\) | Proposition 3.3 (one starter) |
| \([8, 4, 2]\) | Proposition 3.3 (one starter) |
| \([7, 5, 2]\) | Proposition 3.3 (one starter) |
| \([6, 6, 2]\) | Proposition 3.3 (one starter) |
| \([8, 3, 3]\) | Proposition 3.4 (two starters) |
| \([7, 4, 3]\) | Proposition 3.4 (two starters) |
| \([6, 5, 3]\) | Proposition 3.4 (two starters) |
| \([6, 4, 4]\) | Proposition 3.4 (two starters) |
| \([5, 5, 4]\) | Proposition 3.4 (two starters) |
| \([12,2]\) | Proposition 3.3 (one starter) |
| \([11,3]\) | Proposition 3.4 (two starters) |
| \([10,4]\) | Proposition 3.4 (two starters) |
| \([9,5]\) | Proposition 3.4 (two starters) |
| \([8,6]\) | Proposition 3.4 (two starters) |
| \([7,7]\) | Theorem 1.2 |
| \([14]\) | Theorem 1.2 |
Case \(n=15\): see [10, Appendix G] for the computational results.
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([3, 2, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([4, 3, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([3, 3, 3, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([7, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 3, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 4, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 3, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([4, 4, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([4, 3, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| \([3, 3, 3, 3, 3]\) | Theorem 1.2 |
| \([9, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([8, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([7, 4, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 5, 2, 2]\) | Proposition 3.5 (three starters) |
| \([7, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| \([6, 4, 3, 2]\) | Proposition 3.5 (three starters) |
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([5, 5, 3, 2]\) | Proposition 3.5 (three starters) |
| \([5, 4, 4, 2]\) | Proposition 3.5 (three starters) |
| \([6, 3, 3, 3]\) | Theorem 1.2 |
| \([5, 4, 3, 3]\) | Theorem 1.2 |
| \([4, 4, 4, 3]\) | Theorem 1.2 |
| \([11, 2, 2]\) | Proposition 3.5 (three starters) |
| \([10, 3, 2]\) | Proposition 3.5 (three starters) |
| \([9, 4, 2]\) | Proposition 3.5 (three starters) |
| \([8, 5, 2]\) | Proposition 3.5 (three starters) |
| \([7, 6, 2]\) | Proposition 3.5 (three starters) |
| \([9, 3, 3]\) | Theorem 1.2 |
| \([8, 4, 3]\) | Theorem 1.2 |
| \([7, 5, 3]\) | Theorem 1.2 |
| \([6, 6, 3]\) | Theorem 1.2 |
| \([7, 4, 4]\) | Theorem 1.2 |
| \([6, 5, 4]\) | Theorem 1.2 |
| \([5, 5, 5]\) | Theorem 1.2 |
| \([13,2]\) | Theorem 1.2 |
| \([12,3]\) | Theorem 1.2 |
| \([11,4]\) | Theorem 1.2 |
| \([10,5]\) | Theorem 1.2 |
| \([9,6]\) | Theorem 1.2 |
| \([8,7]\) | Theorem 1.2 |
| \([15]\) | Theorem 1.2 |
Case \(n=16\): see [10, Appendix H] for the computational results.
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([2, 2, 2, 2, 2, 2, 2, 2]\) | Theorem 1.2 |
| \([4, 2, 2, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([3, 3, 2, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 2, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([5, 3, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([4, 4, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([4, 3, 3, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([3, 3, 3, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([8, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([7, 3, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 4, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([5, 5, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 3, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([5, 4, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([4, 4, 4, 2, 2]\) | Proposition 3.3 (one starter) |
| \([5, 3, 3, 3, 2]\) | Proposition 3.4 (two starters) |
| \([4, 4, 3, 3, 2]\) | Proposition 3.4 (two starters) |
| \([4, 3, 3, 3, 3]\) | Proposition 3.3 (one starter) |
| \([10, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([9, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([8, 4, 2, 2]\) | Proposition 3.3 (one starter) |
| \([7, 5, 2, 2]\) | Proposition 3.3 (one starter) |
| \([6, 6, 2, 2]\) | Proposition 3.3 (one starter) |
| \([8, 3, 3, 2]\) | Proposition 3.4 (two starters) |
| \([7, 4, 3, 2]\) | Proposition 3.4 (two starters) |
| \([6, 5, 3, 2]\) | Proposition 3.4 (two starters) |
| \([6, 4, 4, 2]\) | Proposition 3.4 (two starters) |
| \([5, 5, 4, 2]\) | Proposition 3.4 (two starters) |
| \([7, 3, 3, 3]\) | Proposition 3.3 (one starter) |
| \([6, 4, 3, 3]\) | Proposition 3.3 (one starter) |
| \([5, 5, 3, 3]\) | Proposition 3.3 (one starter) |
| \([5, 4, 4, 3]\) | Proposition 3.3 (one starter) |
| \([4, 4, 4, 4]\) | Theorem 1.2 |
| \([12, 2, 2]\) | Proposition 3.3 (one starter) |
| \([11, 3, 2]\) | Proposition 3.4 (two starters) |
| \([10, 4, 2]\) | Proposition 3.4 (two starters) |
| \([9, 5, 2]\) | Proposition 3.4 (two starters) |
| \([8, 6, 2]\) | Proposition 3.4 (two starters) |
| \([7, 7, 2]\) | Proposition 3.4 (two starters) |
| \([10, 3, 3]\) | Proposition 3.3 (one starter) |
| \([9, 4, 3]\) | Proposition 3.3 (one starter) |
| \([8, 5, 3]\) | Proposition 3.3 (one starter) |
| \([7, 6, 3]\) | Proposition 3.3 (one starter) |
| \([8, 4, 4]\) | Theorem 1.2 |
| \([7, 5, 4]\) | Proposition 3.3 (one starter) |
| \([6, 6, 4]\) | Proposition 3.3 (one starter) |
| \([6, 5, 5]\) | Proposition 3.3 (one starter) |
| \([14, 2]\) | Proposition 3.4 (two starters) |
| \([13, 3]\) | Proposition 3.3 (one starter) |
| \([12, 4]\) | Theorem 1.2 |
| \([11, 5]\) | Proposition 3.3 (one starter) |
| \([10, 6]\) | Proposition 3.3 (one starter) |
| \([9, 7]\) | Proposition 3.3 (one starter) |
| \([8, 8]\) | Theorem 1.2 |
| \([16]\) | Theorem 1.2 |
Case \(n=17\): see [10, Appendix I] for the computational results.
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([3, 2, 2, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 2, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([4, 3, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([3, 3, 3, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([7, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 3, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 4, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 3, 3, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([4, 4, 3, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([4, 3, 3, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([3, 3, 3, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| \([9, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([8, 3, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([7, 4, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 5, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([7, 3, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 4, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 5, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 4, 4, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 3, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| \([5, 4, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| \([4, 4, 4, 3, 2]\) | Proposition 3.5 (three starters) |
| \([5, 3, 3, 3, 3]\) | Theorem 1.2 |
| \([4, 4, 3, 3, 3]\) | Theorem 1.2 |
| \([11, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([10, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([9, 4, 2, 2]\) | Proposition 3.5 (three starters) |
| \([8, 5, 2, 2]\) | Proposition 3.5 (three starters) |
| \([7, 6, 2, 2]\) | Proposition 3.5 (three starters) |
| \([9, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| \([8, 4, 3, 2]\) | Proposition 3.5 (three starters) |
| \([7, 5, 3, 2]\) | Proposition 3.5 (three starters) |
| \([6, 6, 3, 2]\) | Proposition 3.5 (three starters) |
| \([7, 4, 4, 2]\) | Proposition 3.5 (three starters) |
| \([6, 5, 4, 2]\) | Proposition 3.5 (three starters) |
| \([5, 5, 5, 2]\) | Proposition 3.5 (three starters) |
| \([8, 3, 3, 3]\) | Theorem 1.2 |
| \([7, 4, 3, 3]\) | Theorem 1.2 |
| \([6, 5, 3, 3]\) | Theorem 1.2 |
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([6, 4, 4, 3]\) | Theorem 1.2 |
| \([5, 5, 4, 3]\) | Theorem 1.2 |
| \([5, 4, 4, 4]\) | Theorem 1.2 |
| \([13, 2, 2]\) | Proposition 3.5 (three starters) |
| \([12, 3, 2]\) | Proposition 3.5 (three starters) |
| \([11, 4, 2]\) | Proposition 3.5 (three starters) |
| \([10, 5, 2]\) | Proposition 3.5 (three starters) |
| \([9, 6, 2]\) | Proposition 3.5 (three starters) |
| \([8, 7, 2]\) | Proposition 3.5 (three starters) |
| \([11, 3, 3]\) | Theorem 1.2 |
| \([10, 4, 3]\) | Theorem 1.2 |
| \([9, 5, 3]\) | Theorem 1.2 |
| \([8, 6, 3]\) | Theorem 1.2 |
| \([7, 7, 3]\) | Theorem 1.2 |
| \([9, 4, 4]\) | Theorem 1.2 |
| \([8, 5, 4]\) | Theorem 1.2 |
| \([7, 6, 4]\) | Theorem 1.2 |
| \([7, 5, 5]\) | Theorem 1.2 |
| \([6, 6, 5]\) | Theorem 1.2 |
| \([15, 2]\) | Theorem 1.2 |
| \([14, 3]\) | Theorem 1.2 |
| \([13, 4]\) | Theorem 1.2 |
| \([12, 5]\) | Theorem 1.2 |
| \([11, 6]\) | Theorem 1.2 |
| \([10, 7]\) | Theorem 1.2 |
| \([9, 8]\) | Theorem 1.2 |
| \([17]\) | Theorem 1.2 |
Case \(n=18\): see [10, Appendix J] for the computational results.
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([2, 2, 2, 2, 2, 2, 2, 2, 2]\) | Theorem 1.2 |
| \([4, 2, 2, 2, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([3, 3, 2, 2, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 2, 2, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([5, 3, 2, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([4, 4, 2, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([4, 3, 3, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([3, 3, 3, 3, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([8, 2, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([7, 3, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([6, 4, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([5, 5, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 3, 3, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([5, 4, 3, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([4, 4, 4, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([5, 3, 3, 3, 2, 2]\) | Proposition 3.4(two starters) |
| \([4, 4, 3, 3, 2, 2]\) | Proposition 3.4 (two starters) |
| \([4, 3, 3, 3, 3, 2]\) | Proposition 3.3 (one starter) |
| \([3, 3, 3, 3, 3, 3]\) | Theorem 1.2 |
| \([10, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([9, 3, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([8, 4, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([7, 5, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([6, 6, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([8, 3, 3, 2, 2]\) | Proposition 3.4 (two starters) |
| \([7, 4, 3, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 5, 3, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 4, 4, 2, 2]\) | Proposition 3.4 (two starters) |
| \([5, 5, 4, 2, 2]\) | Proposition 3.4 (two starters) |
| \([7, 3, 3, 3, 2]\) | Proposition 3.3 (one starter) |
| \([6, 4, 3, 3, 2]\) | Proposition 3.3 (one starter) |
| \([5, 5, 3, 3, 2]\) | Proposition 3.3 (one starter) |
| \([5, 4, 4, 3, 2]\) | Proposition 3.3 (one starter) |
| \([4, 4, 4, 4, 2]\) | Proposition 3.3 (one starter) |
| \([6, 3, 3, 3, 3]\) | Proposition 3.4 (two starters) |
| \([5, 4, 3, 3, 3]\) | Proposition 3.4 (two starters) |
| \([4, 4, 4, 3, 3]\) | Proposition 3.4 (two starters) |
| \([12, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([11, 3, 2, 2]\) | Proposition 3.4 (two starters) |
| \([10, 4, 2, 2]\) | Proposition 3.4 (two starters) |
| \([9, 5, 2, 2]\) | Proposition 3.4 (two starters) |
| \([8, 6, 2, 2]\) | Proposition 3.4 (two starters) |
| \([7, 7, 2, 2]\) | Proposition 3.4 (two starters) |
| \([10, 3, 3, 2]\) | Proposition 3.3 (one starter) |
| \([9, 4, 3, 2]\) | Proposition 3.3 (one starter) |
| \([8, 5, 3, 2]\) | Proposition 3.3 (one starter) |
| \([7, 6, 3, 2]\) | Proposition 3.3 (one starter) |
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([8, 4, 4, 2]\) | Proposition 3.3 (one starter) |
| \([7, 5, 4, 2]\) | Proposition 3.3 (one starter) |
| \([6, 6, 4, 2]\) | Proposition 3.3 (one starter) |
| \([6, 5, 5, 2]\) | Proposition 3.3 (one starter) |
| \([9, 3, 3, 3]\) | Proposition 3.4 (two starters) |
| \([8, 4, 3, 3]\) | Proposition 3.4 (two starters) |
| \([7, 5, 3, 3]\) | Proposition 3.4 (two starters) |
| \([6, 6, 3, 3]\) | Proposition 3.4 (two starters) |
| \([7, 4, 4, 3]\) | Proposition 3.4 (two starters) |
| \([6, 5, 4, 3]\) | Proposition 3.4 (two starters) |
| \([5, 5, 5, 3]\) | Proposition 3.4 (two starters) |
| \([6, 4, 4, 4]\) | Proposition 3.4 (two starters) |
| \([5, 5, 4, 4]\) | Proposition 3.4 (two starters) |
| \([14, 2, 2]\) | Proposition 3.4 (two starters) |
| \([13, 3, 2]\) | Proposition 3.3 (one starter) |
| \([12, 4, 2]\) | Proposition 3.3 (one starter) |
| \([11, 5, 2]\) | Proposition 3.3 (one starter) |
| \([10, 6, 2]\) | Proposition 3.3 data-reference-type=”ref” data-reference=”pro:main1″>[pro:main1] (one starter) |
| \([9, 7, 2]\) | Proposition 3.3 (one starter) |
| \([8, 8, 2]\) | Proposition 3.3 (one starter) |
| \([12, 3, 3]\) | Proposition 3.4 (two starters) |
| \([11, 4, 3]\) | Proposition 3.4 (two starters) |
| \([10, 5, 3]\) | Proposition 3.4 (two starters) |
| \([9, 6, 3]\) | Proposition 3.4 (two starters) |
| \([8, 7, 3]\) | Proposition 3.4 (two starters) |
| \([10, 4, 4]\) | Proposition 3.4 (two starters) |
| \([9, 5, 4]\) | Proposition 3.4 (two starters) |
| \([8, 6, 4]\) | Proposition 3.4 (two starters) |
| \([7, 7, 4]\) | Proposition 3.4 (two starters) |
| \([8, 5, 5]\) | Proposition 3.4 (two starters) |
| \([7, 6, 5]\) | Proposition 3.4 (two starters) |
| \([6, 6, 6]\) | Theorem 1.2 |
| \([16, 2]\) | Proposition 3.3 (one starter) |
| \([15, 3]\) | Proposition 3.4 (two starters) |
| \([14, 4]\) | Proposition 3.4 (two starters) |
| \([13, 5]\) | Proposition 3.4 (two starters) |
| \([12, 6]\) | Proposition 3.4 (two starters) |
| \([11, 7]\) | Proposition 3.4 (two starters) |
| \([10, 8]\) | Proposition 3.4 (two starters) |
| \([9, 9]\) | Theorem 1.2 |
| \([18]\) | Theorem 1.2 |
Case \(n=19\): see [10, Appendix K] for the computational results.
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([3, 2, 2, 2, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 2, 2, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([4, 3, 2, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([3, 3, 3, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([7, 2, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 3, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 4, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 3, 3, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([4, 4, 3, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([4, 3, 3, 3, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([3, 3, 3, 3, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([9, 2, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([8, 3, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([7, 4, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 5, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([7, 3, 3, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 4, 3, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 5, 3, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 4, 4, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 3, 3, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 4, 3, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([4, 4, 4, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 3, 3, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| \([4, 4, 3, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| \([4, 3, 3, 3, 3, 3]\) | Theorem 1.2 |
| \([11, 2, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([10, 3, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([9, 4, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([8, 5, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([7, 6, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([9, 3, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([8, 4, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([7, 5, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 6, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([7, 4, 4, 2, 2]\) | Proposition 3.5 (three starters) |
| \([6, 5, 4, 2, 2]\) | Proposition 3.5 (three starters) |
| \([5, 5, 5, 2, 2]\) | Proposition 3.5 (three starters) |
| \([8, 3, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| \([7, 4, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([6, 5, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| \([6, 4, 4, 3, 2]\) | Proposition 3.5 (three starters) |
| \([5, 5, 4, 3, 2]\) | Proposition 3.5 (three starters) |
| \([5, 4, 4, 4, 2]\) | Proposition 3.5 (three starters) |
| \([7, 3, 3, 3, 3]\) | Theorem 1.2 |
| \([6, 4, 3, 3, 3]\) | Theorem 1.2 |
| \([5, 5, 3, 3, 3]\) | Theorem 1.2 |
| \([5, 4, 4, 3, 3]\) | Theorem 1.2 |
| \([4, 4, 4, 4, 3]\) | Theorem 1.2 |
| \([13, 2, 2, 2]\) | Proposition 3.5 (three starters) |
| \([12, 3, 2, 2]\) | Proposition 3.5 (three starters) |
| \([11, 4, 2, 2]\) | Proposition 3.5 (three starters) |
| \([10, 5, 2, 2]\) | Proposition 3.5 (three starters) |
| \([9, 6, 2, 2]\) | Proposition 3.5 (three starters) |
| \([8, 7, 2, 2]\) | Proposition 3.5 (three starters) |
| \([11, 3, 3, 2]\) | Proposition 3.5 (three starters) |
| \([10, 4, 3, 2]\) | Proposition 3.5 (three starters) |
| \([9, 5, 3, 2]\) | Proposition 3.5 (three starters) |
| \([8, 6, 3, 2]\) | Proposition 3.5 (three starters) |
| \([7, 7, 3, 2]\) | Proposition 3.5 (three starters) |
| \([9, 4, 4, 2]\) | Proposition 3.5 (three starters) |
| \([8, 5, 4, 2]\) | Proposition 3.5 (three starters) |
| \([7, 6, 4, 2]\) | Proposition 3.5 (three starters) |
| \([7, 5, 5, 2]\) | Proposition 3.5 (three starters) |
| \([6, 6, 5, 2]\) | Proposition 3.5 (three starters) |
| \([10, 3, 3, 3]\) | Theorem 1.2 |
| \([9, 4, 3, 3]\) | Theorem 1.2 |
| \([8, 5, 3, 3]\) | Theorem 1.2 |
| \([7, 6, 3, 3]\) | Theorem 1.2 |
| \([8, 4, 4, 3]\) | Theorem 1.2 |
| \([7, 5, 4, 3]\) | Theorem 1.2 |
| \([6, 6, 4, 3]\) | Theorem 1.2 |
| \([6, 5, 5, 3]\) | Theorem 1.2 |
| \([7, 4, 4, 4]\) | Theorem 1.2 |
| \([6, 5, 4, 4]\) | Theorem 1.2 |
| \([5, 5, 5, 4]\) | Theorem 1.2 |
| \([15, 2, 2]\) | Proposition 3.5 (three starters) |
| \([14, 3, 2]\) | Proposition 3.5 (three starters) |
| \([13, 4, 2]\) | Proposition 3.5 (three starters) |
| \([12, 5, 2]\) | Proposition 3.5 (three starters) |
| \([11, 6, 2]\) | Proposition 3.5 (three starters) |
| \([10, 7, 2]\) | Proposition 3.5 (three starters) |
| \([9, 8, 2]\) | Proposition 3.5 (three starters) |
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([13, 3, 3]\) | Theorem 1.2 |
| \([12, 4, 3]\) | Theorem 1.2 |
| \([11, 5, 3]\) | Theorem 1.2 |
| \([10, 6, 3]\) | Theorem 1.2 |
| \([9, 7, 3]\) | Theorem 1.2 |
| \([8, 8, 3]\) | Theorem 1.2 |
| \([11, 4, 4]\) | Theorem 1.2 |
| \([10, 5, 4]\) | Theorem 1.2 |
| \([9, 6, 4]\) | Theorem 1.2 |
| \([8, 7, 4]\) | Theorem 1.2 |
| \([9, 5, 5]\) | Theorem 1.2 |
| \([8, 6, 5]\) | Theorem 1.2 |
| \([7, 7, 5]\) | Theorem 1.2 |
| \([7, 6, 6]\) | Theorem 1.2 |
| \([17, 2]\) | Theorem 1.2 |
| \([16, 3]\) | Theorem 1.2 |
| \([15, 4]\) | Theorem 1.2 |
| \([14, 5]\) | Theorem 1.2 |
| \([13, 6]\) | Theorem 1.2 |
| \([12, 7]\) | Theorem 1.2 |
| \([11, 8]\) | Theorem 1.2 |
| \([10, 9]\) | Theorem 1.2 |
| \([19]\) | Theorem 1.2 |
Case \(n=20\): see [10, Appendix L] for the computational results.
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([2, 2, 2, 2, 2, 2, 2, 2, 2, 2]\) | Theorem 1.2 |
| \([4, 2, 2, 2, 2, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([3, 3, 2, 2, 2, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 2, 2, 2, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([5, 3, 2, 2, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([4, 4, 2, 2, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([4, 3, 3, 2, 2, 2, 2, 2]\) | Proposition 3.3 (two starters) |
| \([3, 3, 3, 3, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([8, 2, 2, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([7, 3, 2, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 4, 2, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([5, 5, 2, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 3, 3, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([5, 4, 3, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([4, 4, 4, 2, 2, 2, 2]\) | Proposition 3.3 | (one starter)
| \([5, 3, 3, 3, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([4, 4, 3, 3, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([4, 3, 3, 3, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([3, 3, 3, 3, 3, 3, 2]\) | Proposition 3.4 (two starters) |
| \([10, 2, 2, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([9, 3, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([8, 4, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([7, 5, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([6, 6, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([8, 3, 3, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([7, 4, 3, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 5, 3, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([6, 4, 4, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([5, 5, 4, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([7, 3, 3, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([6, 4, 3, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([5, 5, 3, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([5, 4, 4, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([4, 4, 4, 4, 2, 2]\) | Proposition 3.3 (one starter) |
| \([6, 3, 3, 3, 3, 2]\) | Proposition 3.4 (two starters) |
| \([5, 4, 3, 3, 3, 2]\) | Proposition 3.4 (two starters) |
| \([4, 4, 4, 3, 3, 2]\) | Proposition 3.4 (two starters) |
| \([5, 3, 3, 3, 3, 3]\) | Proposition 3.3 (one starter) |
| \([4, 4, 3, 3, 3, 3]\) | Proposition 3.3 (one starter) |
| \([12, 2, 2, 2, 2]\) | Proposition 3.3 (one starter) |
| \([11, 3, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([10, 4, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([9, 5, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([8, 6, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([7, 7, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([10, 3, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([9, 4, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([8, 5, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([7, 6, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([8, 4, 4, 2, 2]\) | Proposition 3.3 (one starter) |
| \([7, 5, 4, 2, 2]\) | Proposition 3.3 (one starter) |
| \([6, 6, 4, 2, 2]\) | Proposition 3.3 (one starter) |
| \([6, 5, 5, 2, 2]\) | Proposition 3.3 (one starter) |
| \([9, 3, 3, 3, 2]\) | Proposition 3.4 (two starters) |
| \([8, 4, 3, 3, 2]\) | Proposition 3.4 (two starters) |
| \([7, 5, 3, 3, 2]\) | Proposition 3.4 (two starters) |
| \([6, 6, 3, 3, 2]\) | Proposition 3.4 (two starters) |
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([7, 4, 4, 3, 2]\) | Proposition 3.4 (two starters) |
| \([6, 5, 4, 3, 2]\) | Proposition 3.4 (two starters) |
| \([5, 5, 5, 3, 2]\) | Proposition 3.4 (two starters) |
| \([6, 4, 4, 4, 2]\) | Proposition 3.4 (two starters) |
| \([5, 5, 4, 4, 2]\) | Proposition 3.4 (two starters) |
| \([8, 3, 3, 3, 3]\) | Proposition 3.3 (one starter) |
| \([7, 4, 3, 3, 3]\) | Proposition 3.3 (one starter) |
| \([6, 5, 3, 3, 3]\) | Proposition 3.3 (one starter) |
| \([6, 4, 4, 3, 3]\) | Proposition 3.3 (one starter) |
| \([5, 5, 4, 3, 3]\) | Proposition 3.3 (one starter) |
| \([5, 4, 4, 4, 3]\) | Proposition 3.3 (one starter) |
| \([4, 4, 4, 4, 4]\) | Theorem 1.2 |
| \([14, 2, 2, 2]\) | Proposition 3.4 (two starters) |
| \([13, 3, 2, 2]\) | Proposition 3.3 (one starter) |
| \([12, 4, 2, 2]\) | Proposition 3.3 (one starter) |
| \([11, 5, 2, 2]\) | Proposition 3.3 (one starter) |
| \([10, 6, 2, 2]\) | Proposition 3.3 (one starter) |
| \([9, 7, 2, 2]\) | Proposition 3.3 (one starter) |
| \([8, 8, 2, 2]\) | Proposition 3.3 (one starter) |
| \([12, 3, 3, 2]\) | Proposition 3.4 (two starters) |
| \([11, 4, 3, 2]\) | Proposition 3.4 (two starters) |
| \([10, 5, 3, 2]\) | Proposition 3.4 (two starters) |
| \([9, 6, 3, 2]\) | Proposition 3.4 (two starters) |
| \([8, 7, 3, 2]\) | Proposition 3.4 (two starters) |
| \([10, 4, 4, 2]\) | Proposition 3.4 (two starters) |
| \([9, 5, 4, 2]\) | Proposition 3.4 (two starters) |
| \([8, 6, 4, 2]\) | Proposition 3.4 (two starters) |
| \([7, 7, 4, 2]\) | Proposition 3.4 (two starters) |
| \([8, 5, 5, 2]\) | Proposition 3.4 (two starters) |
| \([7, 6, 5, 2]\) | Proposition 3.4 (two starters) |
| \([6, 6, 6, 2]\) | Proposition 3.4 (two starters) |
| \([11, 3, 3, 3]\) | Proposition 3.3 (one starter) |
| \([10, 4, 3, 3]\) | Proposition 3.3 (one starter) |
| \([9, 5, 3, 3]\) | Proposition 3.3 (one starter) |
| \([8, 6, 3, 3]\) | Proposition 3.3 (one starter) |
| \([7, 7, 3, 3]\) | Proposition 3.3 (one starter) |
| \([9, 4, 4, 3]\) | Proposition 3.3 (one starter) |
| \([8, 5, 4, 3]\) | Proposition 3.3 (one starter) |
| \([7, 6, 4, 3]\) | Proposition 3.3 (one starter) |
| \([7, 5, 5, 3]\) | Proposition 3.3 (one starter) |
| \([6, 6, 5, 3]\) | Proposition 3.3 (one starter) |
| \([8, 4, 4, 4]\) | Theorem 3.3 |
| 2-factor type \([m_1,\ldots,m_t]\) | HOP\((2m_1,\ldots,2m_t)\) has a solution by… |
|---|---|
| \([7, 5, 4, 4]\) | Proposition 3.3 (one starter) |
| \([6, 6, 4, 4]\) | Proposition 3.3 (one starter) |
| \([6, 5, 5, 4]\) | Proposition 3.3 (one starter) |
| \([5, 5, 5, 5]\) | Theorem 1.2 |
| \([16, 2, 2]\) | Proposition 3.3 (one starter) |
| \([15, 3, 2]\) | Proposition 3.4 (two starters) |
| \([14, 4, 2]\) | Proposition 3.4 (two starters) |
| \([13, 5, 2]\) | Proposition 3.4 (two starters) |
| \([12, 6, 2]\) | Proposition 3.4 (two starters) |
| \([11, 7, 2]\) | Proposition 3.4 (two starters) |
| \([10, 8, 2]\) | Proposition 3.4 (two starters) |
| \([9, 9, 2]\) | Proposition 3.4 (two starters) |
| \([14, 3, 3]\) | Proposition 3.3 (one starter) |
| \([13, 4, 3]\) | Proposition 3.3 (one starter) |
| \([12, 5, 3]\) | Proposition 3.3 (one starter) |
| \([11, 6, 3]\) | Proposition 3.3 (one starter) |
| \([10, 7, 3]\) | Proposition 3.3 (one starter) |
| \([9, 8, 3]\) | Proposition 3.3 (one starter) |
| \([12, 4, 4]\) | Theorem 1.2 |
| \([11, 5, 4]\) | Proposition 3.3 (one starter) |
| \([10, 6, 4]\) | Proposition 3.3 (one starter) |
| \([9, 7, 4]\) | Proposition 3.3 (one starter) |
| \([8, 8, 4]\) | Theorem 1.2 |
| \([10, 5, 5]\) | Proposition 3.3 (one starter) |
| \([9, 6, 5]\) | Proposition 3.3 (one starter) |
| \([8, 7, 5]\) | Proposition 3.3 (one starter) |
| \([8, 6, 6]\) | Proposition 3.3 (one starter) |
| \([7, 7, 6]\) | Proposition 3.3 (one starter) |
| \([18, 2]\) | Proposition 3.4 (two starters) |
| \([17, 3]\) | Proposition 3.3 (one starter) |
| \([16, 4]\) | Theorem 1.2 |
| \([15, 5]\) | Proposition 3.3 (one starter) |
| \([14, 6]\) | Proposition 3.3 (one starter) |
| \([13, 7]\) | Proposition 3.3 (one starter) |
| \([12, 8]\) | Theorem 1.2 |
| \([11, 9]\) | Proposition 3.3 (one starter) |
| \([10, 10]\) | Theorem 1.2 |
| \([20]\) | Theorem 1.2 |
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