In our proof of Theorem 1.10 we construct the loose Hamilton cycle by finding several paths and joining them into a spanning cycle. Here ak-graph P is a path if its vertices can be given
a linear ordering such that every edge ofP consists of k consecutive vertices, and so that every
pair of consecutive vertices of P lie in an edge of P . Similarly as for cycles, we say that a
pathP is loose if edges of P intersect in at most one vertex. The ordering of the vertices of P
naturally gives an ordering of the edges of P . We say that any vertex of P which lies in the
initial edge ofP , but not the second edge of P , is an initial vertex. Similarly, any vertex of P
which lies in the final edge ofP but not the penultimate edge is a final vertex. Also, we refer to
vertices ofP which lie in more than one edge of P as link vertices. So a loose path P has k− 1
initial vertices,k− 1 final vertices, and one link vertex in each pair of consecutive edges.
In Section 4.3, we introduce various ideas we need in the proof of Theorem 1.10. In particu- lar, we state an analogue of the Szemer´edi regularity lemma for hypergraphs due to R¨odl and Schacht [41] and Theorem 4.5 due to Keevash [23]. The latter provides a useful way of applying the hypergraph blow-up lemma. In Section 4.4, we prove various auxiliary results, including a
result on finding loose paths in completek-partite k-graphs, and an approximate minimum de-
gree condition to guarantee a near-perfect packing ofH with a particular k-graph Ak. Finally, in Section 4.5 we use the regularity method to prove Theorem 1.10 as follows.
4.2.1
Imposing structure on
H
We begin in Section 4.5.1 with the following steps, which correspond to those in the description of the regularity method in the introduction.
1. We apply the hypergraph regularity lemma to partition the vertex set ofH into clusters.
2. Next, we define a suitable ‘reducedk-graph’ R of H, as discussed in the introduction.
3. We find copies of a suitable auxiliaryk-graphAkcovering almost all vertices ofR.
We use this structure of the reducedk-graph R to find a Hamilton cycle in H. In the remaining
part of Section 4.5.1, we split the sub-k-graph of H corresponding to each copy of Ak in R into the same number of vertex-disjoint k-partite k-graphs Hi on vertex sets Xi. These are suitable for embedding almost spanning loose paths (the sizes of the vertex classes of eachHi are chosen to meet this condition). We also form an ‘exceptional’ loose pathLewhich contains all of the vertices ofH not contained in any of the Xi (actually, if|V (H)| is not divisible by
k− 1, then Le contains two consecutive edges which intersect in more than one vertex).
4.2.2
The linking strategy
To complete the proof, in Section 4.5.3 we use the structure imposed onH to find a Hamilton
(a) Thek-graphs Hiare connected by means of a walkW = e
1, . . . , e`in the ‘supplementary graph’. This graph (which we define in Section 4.5.2) has vertices1, . . . , t0corresponding to thek-graphs Hi.
(b) Using Lemma 4.19, each edgeej ofW is used to create a short ‘connecting’ loose path
Lj inH joining two different His.
(c) Le and the pathsLj are extended to ‘prepaths’ (these can be thought of as a path minus an initial vertex and a final vertex)L∗
e = I0LeF0 andL∗j = IjLjFj, whereI0, F0 and all
Ij, Fj are sets of sizek− 2. These prepaths have the property that there are large sets Ij0 andF0
j such thatL∗j can be extended to a loose path by adding any vertex ofIj0 as an initial vertex and any vertex ofF0
j as a final vertex. Similarly there are large setsI`+10 andF00 so thatL∗e can be extended to a path by adding any vertex ofI`+10 as an initial vertex and any vertex ofF00 as a final vertex. Ij+10 andFj0 both lie in the sameHi(for allj = 0, . . . , `). (d) For eachHiand for all those pairsI0
j+1, Fj0 which lie inHi, we choose a loose pathL0j+1 insideHifromFj0toIj+10 . For eachi, we use the hypergraph blow-up lemma (in the form of Theorem 4.5) to ensure that together all thoseL0
j which lie inHiuse all the remaining vertices ofHi.
(e) The loose Hamilton cycle is then the concatenationL∗eL01L∗1. . . L0`L∗`L0`+1.
4.2.3
Controlling divisibility
Note that the number of vertices of a loose path is 1 modulo k − 1. So in order to apply
Theorem 4.5 to obtain spanning loose paths in a subgraph of Hi, we need this subgraph to satisfy this condition. So we choose our paths sequentially to satisfy the following congruences modulok− 1.
(b) LetXi(j−1) be the subset of Xiobtained by removingV (L
1), . . . , V (Lj−1). (All the Xi are disjoint fromV (Le).) Let dibe the number of times thatW visits Hi. When choosing
Lj, for everyXi it traverses (except the final one) we arrange to intersectXi(j− 1) in a set of size≡ ti(j)≡ |Xi(j−1)| + di(the size modulok−1 of the intersection of Ljwith the final Xi it traverses is then determined by the sizes of the other intersections). The choice ofLein (a) ensures that after allLj have been picked, the remaining partXi(`) of
Xihas size≡ −d i.
(c) EachLj is extended to a prepathL∗j by addingIj andFj. Similarly,Le is extended into a prepathL∗
eby addingI0andF0. Now the remaining part ofXihas size≡ di.
(d) It remains to selectdipathsL0j within eachXi: each uses≡ 1 vertices, so the divisibility conditions are satisfied.