4.5 Proof of Theorem 1.10
4.5.3 Constructing the loose Hamilton cycle
Our Hamilton cycle ofH is to consist of Le and paths in eachHi as well as paths connecting the Xi. However, we need to make sure that all these paths join up nicely, motivating the following definition. Suppose L is a path in some k-graph K with initial vertex x0 and final vertexy0. Also, letI, F ⊆ V (K) \ V (L) be disjoint sets of size k − 2. Then L∗ = I∪ F ∪ V (L) is a prepath. Note that L∗ is not (the vertex set of) a k-graph, but that if we can find vertices
x, y ∈ V (K) \ L∗ such that{x, x0} ∪ I, {y, y0} ∪ F ∈ K, then adding x and y to L∗ gives another path. We refer to all such verticesx ∈ V (K) as possible initial vertices of L∗ and to all such verticesy ∈ V (K) as possible final vertices. If L, L0 andL00 are disjoint loose paths,
I, F, x, y are as before, x is also the final vertex of L0andy is also the initial vertex of L00thenI
andF together with L0, L, L00form a single loose path, illustrating how we join paths together.
We start by converting our exceptional pathLe into a prepath. Recall that|V (Le)| < ηn and that the initial vertexxe ofLe and its final vertexye satisfy (C6). Leta ∈ [t0] and ua ∈ [k] be such thatxe ∈ Xuaa. Pick anyu
0
a ∈ [k] with ua 6= u0a. (C4) and (C6) together imply that there is a setI0 ⊆ Xa\ (Xuaa ∪ X
a
u0a) for which Xua0a contains at leastc|Xa| vertices v which form an
edge ofHatogether withI
0∪ {xe}. Let I00 ⊆ Xua0a be such a set of vertices. Similarly, letting
b ∈ [t0], u
b 6= u0b ∈ [k] be such that ye ∈ Xubb, there is a setF0 ⊆ X
b \ (Xb ub ∪ X b u0 b ∪ I0) for whichXb u0 bcontains at leastc|X
b| vertices v which form an edge of Hbtogether withF
0∪ {ye}. LetF0
0 ⊆ Xub0
b be such a set of vertices. LetL
∗
e be the prepathI0∪ F0∪ V (Le). Then I00 is a set of possible initial vertices ofL∗
e andF00 is a set of possible final vertices. (We do not removeI0 fromXaandF
0fromXbat this stage.)
Since by Lemma 4.18 the supplementary graphR∗ is connected, we can find a walkW from b toa in R∗such that everyi∈ [t0] = V (R∗) appears as an initial, link or final vertex in W (these vertices were defined in Section 4.4.2) and such thatW has length ` 6 (t0)2. Let e
the edges of this walk, letr1 = b, r`+1 = a, and let r2, . . . , r` be the link vertices of the walk. For eachi∈ [t0], let d
i =|{j ∈ [` + 1] : rj = i}|, i.e. the number of times i appears as an initial, link or final vertex inW . So di > 0 for every i andPdi = ` + 1.
Next we apply Lemma 4.19 to each edgeejto find a loose pathLjinH. We then extend Lj to a prepathL∗j with many possible initial vertices inXrj and many possible final vertices inXrj+1.
We do this for eache1, . . . , e` in turn. So suppose thats ∈ [`] and that for all j = 1, . . . , s − 1 we have defined loose paths Lj inH as well as sets Ij, Fj extendingLj to a prepathL∗j which satisfy the following properties:
(D1) Lj lies in the sub-k-graph of H induced by
S
i∈ejX
i and contains at most4k3vertices.
(D2) The initial vertexxj ofLj lies inXrj and its final vertexyj lies inXrj+1. (D3) Ij ⊆ Xrj andFj ⊆ Xrj+1.
(D4) There is a setIj0 ⊆ Xrj of at leastc|Xrj| possible initial vertices for L∗j. Similarly, there is a setFj0 ⊆ Xrj+1 of at leastc|Xrj+1| possible final vertices for L∗j.
(D5) All the prepathsL∗
e, L∗1, . . . , L∗s−1 are disjoint.
(D6) For each i ∈ [t0] and all j = 0, . . . , s− 1 let Xi(j) = Xi \ (V (L
1)∪ · · · ∪ V (Lj)), where Xi(0) = Xi. For each j ∈ [s − 1] set t
i(j) = |Xi(j − 1)| + di. Then for every i ∈ ej with i 6= rj+1 we have |V (Lj) ∩ Xi| ≡ ti(j) mod k − 1. Moreover
|V (Lj)∩ Xrj+1| ≡ 1 −Pi∈ej, i6=rj+1ti(j) mod k− 1.
Let us now show how to findLs,IsandFs. Apply Lemma 4.19 withe = es,i0 = rs,i00 = rs+1 and with Z = L∗
1 ∪ . . . L∗s−1 ∪ I0 ∪ F0 to find a loose path Ls which satisfies (D1), (D2), (D6) and is disjoint fromL∗e, L∗1, . . . , L∗s−1. Moreover, the initial vertexxsofLslies in at least
|Hrs|/(2|Xrs|) edges of Hrs, and the final vertex y
edges ofHrs+1. We can now use the latter property to choose sets I
s and Fs which extendLs to a prepathL∗
s satisfying (D3)–(D5). The argument for this is similar to that for the extension ofLetoL∗e. Altogether this shows that we can find prepathsL∗1, . . . , L∗` satisfying (D1)–(D6).
For eachi ∈ [t0] we let j
i be the maximal integer such thati ∈ eji. Thus X
i(`) = Xi(j i) =
Xi(j
i− 1) \ V (Lji) by (D1). But if i6= r`+1 then (D5) and (D6) together imply that
|V (Lji)∩ X
i(j
i− 1)| = |V (Lji)∩ X
i
| ≡ ti(ji)≡ |Xi(ji− 1)| + di mod k− 1 and so|Xi(`)| ≡ −di mod k− 1. We claim that this also holds if i = r`+1. To see this, recall that sinceLj is loose, we have|V (Lj)| ≡ 1 mod k − 1 for each j ∈ [`]. Hence
|Xr`+1(`)| = |V (H) \ V (L e)| − X j∈[`] |V (Lj)| − X i∈[t0], i6=r`+1 |Xi(`)| (4.14) ≡ −1 − ` + X i∈[t0] i6=r`+1 di ≡ −dr`+1 mod k− 1
as` + 1 =Pi∈[t0]di. LetYi = Xi\ (L∗e∪ L∗1∪ · · · ∪ L∗`). Since by (D3) for each i ∈ [t0] there
are exactly2(k− 2)divertices ofXi which lie inL∗e, L∗1, . . . , L∗` but not inLe, L1, . . . , L`, this in turn implies that
|Yi| ≡ −di − 2(k − 2)di≡ di mod k− 1. (4.20)
Let x`+1 = xe, y0 = ye, L∗0 = L∗e, I`+1 = I0 and I`+10 = I00. In order to complete our prepathsL∗
0, . . . , L∗` to a Hamilton cycle we wish to choosedi disjoint loose pathsLi1, . . . , Lidi
within eachH[Yi] which together contain all the vertices in Yi and which ‘connect’ successive prepathsL∗
j. We achieve this as follows. Let Ji be the set of allj ∈ [` + 1] with rj = i. So
Ji is the set of positions at whichi occurs as an initial, final or link vertex in our walk W and
|Ji| = di. Let j1 6 . . . 6 jdi be the elements of Ji. Then we choose theL
i
s (s ∈ [di]) in such a way that the initial vertex ofLislies inFj0s−1 and its final vertex lies inI
0
js, all theL
i sare disjoint and together they cover all the vertices inYi. To see that this can be done, first note
that|Xi\ Yi| 6 `(4k3+ 2(k− 2)) + 2(k − 2) εn1. So using Lemma 4.7 together with (C5) and (4.20) it is easy to check that the complete k-partite k-graph on Yi contains such paths (e.g. first chooseLi
1, . . . , Lidi−1, each consisting of precisely 2 edges, and then apply (C5) and
Lemma 4.7 to find a loose pathLi
scontaining all the remaining vertices ofYi). Now (C3) and (D4) together imply thatGi[Yi]\ Mi[Yi] contains the k-complexes induced by these paths (i.e. it contains(Li
1)6, . . . , (Lidi)
6). But this means that we can find the required pathsLi
1, . . . , Lidi
in eachH[Yi].
Finally, for eachs∈ [di] write L0js forL
i
sandx0js for its initial andy
0
js for its final vertex (where
js is as defined in the previous paragraph). To obtain our Hamilton cycle ofH we first traverse
L0 = Le, then we use the edge F0 ∪ {y0, x01} in order to move to the initial vertex x01 of L01. (This is possible sincex0
1 ∈ F00.) Now we traverseL01 and use the edgeI1∪ {y10, x1} to get to x1. (Again, this is possible since y0
1 ∈ I10.) Next we traverse L1 and use the edgeF1 ∪ {y1, x02} to move to x02. We continue in this way until we have reached the initial vertexx`+1 = xe of
L0 = Le again. (So in the last step we traversedL0`+1 and used the edgeI`+1 ∪ {y`+10 , x`+1}.)
CHAPTER
5
H
AMILTON
`-
CYCLES IN UNIFORM
HYPERGRAPHS
In this chapter we prove Theorem 1.11, which is restated below for ease of reference.
Theorem 1.11For allk > 3,1 6 ` 6 k−1such that(k−`) - kand anyη > 0there existsn0so
that ifn > n0and(k− `)|nthen anyk-graphHonnvertices withδ(H) >
1 d k k−`e(k−`) + η n
contains a Hamilton`-cycle.
5.1
Outline of the proof of Theorem 1.11
In our proof of Theorem 1.11 we construct the Hamilton `-cycle by finding several `-paths
and joining them into a spanning`-cycle. Here a k-graph P is an `-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 edges ofP (in the natural ordering induced on the edges) intersect in
precisely` vertices. We say that an enumeration v1, v2, . . . , vr of the vertices ofP is a vertex
We say that ordered setsA and B are ordered ends of P if|A| = |B| = ` and A and B are initial
and final segments of a vertex sequence ofP . This allows us to join up `-paths in the following
manner. Let P and Q be `-paths, and let Pbeg andPend be ordered ends ofP , and Qbeg and
Qend be ordered ends ofQ. Suppose that Pend = Qbeg, and thatV (P )∩ V (Q) = Pend. Then thek-graph with vertex set V (P )∪ V (Q) and with all the edges of P and of Q is an `-path with
ordered endsPbeg andQend.
Our proof of Theorem 1.11 uses ideas of [16], which in turn were based on the ‘absorbing path’ method of [39] and [40]. Our proof contains further developments of the method, which may be of independent interest. We use the regularity method to prove three preliminary lemmas, which are as follows.
• The ‘absorbing path lemma’ states that if (k −`) - k then in any sufficiently large k-graph
of large minimum degree there exists an`-path P which can ‘absorb’ any small set X of
vertices outsideP . By this we mean that for any such small set X there is another `-path Q with the same ordered ends as P and with V (Q) = V (P )∪ X. Then we can think of
replacingP with Q as ‘absorbing’ the vertices of X into P .
• The ‘path cover lemma’ states that any sufficiently large k-graph satisfying the minimum
degree condition of Theorem 1.11 can be almost covered by a bounded number of disjoint
`-paths.
• The ‘diameter lemma’ states that if (k − `) - k then any sufficiently large k-graph of
large minimum degree has small diameter in the sense that we can find an `-path from
any ordered`-set of vertices to any other ordered `-set of vertices.
These three lemmas then combine to prove Theorem 1.11 as follows. Firstly, we find inH an absorbing`-path, and then we almost cover the induced k-graph on the remaining vertices by
`-cycle C which thus contains almost every vertex ofH. Finally, we absorb all vertices of H
not contained inC into our absorbing path, thereby forming an `-cycle containing every vertex
ofH.
The combination of these three preliminary lemmas to prove Theorem 1.11 is very similar to the method of H`an and Schacht in [16]. On the other hand, to prove these preliminary results substantial changes to the method of H`an and Schacht were required. For example, for
1 6 ` < k/2 the diameter lemma as described above is an immediate consequence of the
minimum degree condition ofH. Indeed, if A and B are ordered `-sets of vertices of H, then we may add anyk− 1 − 2` vertices from outside A and B to A ∪ B to obtain a set S of size k− 1. Then we can apply the minimum degree condition of H to find a vertex x ∈ H − S such
thatS∪{x} is an edge of H; then this single edge is the desired path in H. However, if ` > k/2
then things are more difficult. Indeed, in Section 5.4 we use the regularity method (with the notion of strong hypergraph regularity introduced in Section 5.3) to prove a ‘diameter lemma’ as stated above. A similar assertion for the case` = k− 1, (called the ‘Connecting Lemma’),
was proved in [40]. The proof is quite different from ours.
In a similar way, it is more difficult to prove an absorbing path lemma for ` > k/2 than for 1 6 ` < k/2. So whereas H`an and Schacht were able to prove a similar result using only weak
hypergraph regularity, we have found it necessary to use strong hypergraph regularity for the proof of our absorbing path lemma, which is given in Section 5.5. Actually, we cannot absorb arbitrary sets of vertices, but only ‘good’`-sets of vertices. We show that most `-sets of vertices
are good, which is sufficient for our purposes. This weaker notion of absorption may be useful for other problems.
In Section 5.6 we prove the path cover lemma. A similar result was already proved in [40]. The main difference is that they used weak regularity, whereas we have used strong regularity, but this is simply to avoid having to introduce multiple notions of regularity — weak regularity
would have sufficed for this part of our proof.
Finally, in Section 5.7 we complete the proof as outlined earlier.