3.5 Applications
First we obtain the following two corollaries.
Corollary 3.6. Letr, ∆ ≥ 2be integers and H is anr-uniform hypergraph withnvertices,∆(H) ≤ ∆, e(H)> n/randγ(H) =e(H)/(n−2). Then forp=ω n−1/γ(H)
the random graphH(r)(n, p)contains a copy ofH a.a.s., while for everyε >0we have forp≤(1−ε)(e/n)1/γthatP[H ⊆ H(r)(n, p)]→0.
30
3. Riordan’s theorem for hypergraphs
Proof. Since∆is fixed andγ(H) ≤(1 +o(1))∆, condition (2.1) is satisfied. Moreover, e(H) > n/r implies∆(H)≥2. Theorem 2.5 yields the first part of the claim.
LetXbe the number of copies ofHinH(r)(n, p)and we estimate its expectationE[X]as follows:
E[X]≤n!pe(H)≤3√
n(1−ε)e(H)(n/e)2=o(1).
Now Markov’s inequalityP[X ≥1]≤E[X]yields the second part of the corollary.
We call a hypergraphH d-regular if every vertex ofH has degreed.
Corollary 3.7. Letr ≥2be an integer andH be an∆-regularr-uniform hypergraph, where∆ = o(n1/4) but∆ =ω(log1−1/rn). Then for everyε >0we have thatH(r)(n, p)contains a.a.s.Hifp= (1 +ε)n−r/∆. FurthermoreP[H ⊆ H(r)(n, p)]→0forp≤n−r/∆, i.e.p=n−r/∆is a sharp threshold for the appearance of copies ofH inH(r)(n, p).
Proof. LetX count the copies ofH inH(r)(n, p)and forp≤n−r/∆we have
P[X≥1]≤E[X]≤n!n−re(H)/∆=n!n−n =o(1).
Next we boundγ(H)as follows: ∆/r ≤γ(H)≤ ∆r(∆(∆1/(r−1)1/(r−1)+1)
−1). This is obtained from the estimate eH(v)≤min{∆v/r, vr
}by considering two cases whetherv≤∆1/(r−1)+ 1or not. Letε∈(0,1), then
n
(1 +ε)n−r/∆γ(H)
∆−4≥
(1 +ε)n1/γ(H)−r/∆∆−4r(1+o(1))/∆γ(H)
≥
(1 +ε)n−2r/(∆1+1/(r−1)) 1 +o(1)γ(H)
→ ∞,
holds and therefore Theorem 2.5 is applicable and the statement follows.
Thus, Theorem 2.5 (Corollaries 3.6 and 3.7) states that under some technical conditions the thresh-old for the appearance of the spanning structure comes from the expectation threshthresh-oldpE. In the following we derive asymptotically optimal thresholds for the appearance of various spanning struc-tures inH(r)(n, p)which are consequences of the Corollaries 3.6 and 3.7.
Hamilton Cycles
The following is a slightly weaker version of Dudek and Frieze [44]. Recall that anr-uniform hyper-graph is`-Hamiltonian if it containsn/(r−`)edges which form consecutive segments of some cyclic ordering of all vertices and two consecutive edges overlap in`vertices.
Corollary 3.8. For all integersr > `≥2,(r−`)|nandp=ω(n`−r)the random hypergraphH(r)(n, p)is
`-Hamiltonian a.a.s.
Proof. Denote byCn(r,`) an`-overlapping Hamilton cycle onnvertices. It is not difficult to see that γ(Cn(r,`)) = (r−`)(n−2)n . Indeed, letV ⊆[n]be a set of sizev < n. ThenCn(r,`)[V]is a union of vertex-disjoint`-overlapping paths, where an`-overlapping path of lengthsconsists ofs(r−`) +`ordered vertices and edges are consecutive segments intersecting in`vertices. This givese(Cn(r,`)[V])≤(v−
3.5 Applications
`)/(r−`)and from (r−`)(v−2)v−` ≤ (r−`)(n−2)n we get that the optimal value forγ(Cn(r,`))is obtained by the whole cycle, i.e.γ(Cn(r,`)) =(r−`)(n−2)n .
Sincee(Cn(r,`))> n/r,∆(Cn(r,`)) =dr−`r eandn2(r−`)/n→1, Corollary 3.6 implies the statement.
Cube-hypergraphs
Ther-uniformd-dimensional cube-hypergraphQ(r)(d)was studied in [33] and its vertex set isV :=
[r]dand its hyperedges arer-sets of the vertex setV that all differ in one coordinate. Thus,Q(r)(d)has rdvertices,drd−1edges and isd-regular. In the caser = 2this is the usual definition of the (graph) hypercube. The following corollary is a direct consequence of Corollary 3.7.
Corollary 3.9. For all integersr≥2,ε >0andp=r−r+εit holdsP[Q(r)(d)⊆ H(r)(rd, p)]tends to1as dtends to infinity. On the other hand,P[Q(r)(d)⊆ H(r)(rd, r−r)]→0asd→ ∞.
We remark that, in the caser= 2, Riordan [97] proved even better dependence ofεond, and similar dependence can be shown forr >2.
Lattices
Another example considered in [97] was the graph of the lattice Lk, whose vertex set is [k]2 and two vertices are adjacent if their Euclidean distance is one. There it is shown thatp = n−1/2 is asymptotically the threshold. One can viewLkas the cubesQ(2)(2)(these are cyclesC4) gluedalong the edges.
We define the `-overlapping hyperlattice L(r)(`, k) as the r-uniform hypergraph on the vertex set[(k−2)(r−`) +r]2 and the hyperedges being either of the form {(x, i), . . . ,(x, i+r−1)} or {(j, y), . . . ,(j+r−1, y)}, wherex,y∈[(k−2)(r−`) +r]andi,j≡1 mod (r−`). This hypergraph thus arises if we glue together(k−1)2 copies ofQ(r)(2)that overlap on`hyperedges accordingly.
Thus,L(2)(1, k)is just the usual graph latticeLk.
Corollary 3.10. Letr ≥2andkbe an integer. Forp=ω n−1/2
(wheren = (k−2 +r)2) the random r-uniform hypergraphH(r)(n, p)contains a copy ofL(r)(r−1, k)a.a.s. Moreover, forp=n−1/2,P[L(r)(r− 1, k)⊆ H(r)(n, p)]→0ask(and thusn) tends to infinity.
Proof. Observe thatL:=L(r)(r−1, k)has(k−2+r)2vertices (which can be associated with[k−2+r]2) and2(k−1)(k−2 +r)edges.
We aim to show that eL(v) ≤ 2(v−r)for allv ≥ r+ 1. We argue similarly as in [97]. Observe thateL(v)≤2forv =r+ 1. Let nowL0be an arbitrary subhypergraph ofLonv+ 1≤(k−2 +r)2 vertices such thate(L0) =eL(v+ 1). It is easy to see that there is a vertex of degree2inL0(take(i, j) such that(i+ 1, j),(i, j+ 1)6∈V(L0)). It follows that theneL(v+ 1)≤eL(v) + 2forv > r+ 1giving eL(v)≤2(v−r)for allv≥r+ 1.
It follows thatγ := γ(L) ≤ 2 and applying Theorem 2.5 withnpγ = ω(1) yields the first part.
Markov’s inequality yields the second part.
32
3. Riordan’s theorem for hypergraphs
Spheres
Letr ≥3and letGbe a connected planar graph onnvertices with a drawing all of whose faces are cycles of lengthr. We define a sphereSrnas anr-uniform hypergraph all of whose edges correspond to the faces of that particular drawing (note that a sphere is not unique) and the vertex set beingV(G).
Since every edge ofGlies in2 faces and there areredges in every face, we obtain2e(G) = rf(G), where f(G)is the number of faces of G. We thus get from Euler’s formula for planar graphs the condition2v(Snr)−4 = (r−2)e(Snr).
Corollary 3.11. Letr≥3andSbe some sphereSnrwith∆ = ∆(Snr). Then forp=ω ∆2r−4n−(r−2)/2 the randomr-uniform hypergraphH(r)(n, p)contains a copy ofSnra.a.s.
Proof. Let G be a planar graph, out of which the sphereSnr arose. Let V ⊆ V(Snr)and v = |V|. We can assume thatG0 := G[V]is connected27. Therefore, we get from Euler’s formula: f(G0) = 2 +e(G0)−v. On the other hand, by counting edge-face incidences we get: rf(G0) ≤ 2e(G0)and we obtain: (r−2)f(G0) ≤ 2v −4, which yields an upper boundeSr
n(v) ≤ 2v−4r−2. It follows that γ(Srn)≤2/(r−2)holds and this upper bound is attained by the sphereSnritself. Thus, we immediately getγ(Snr) = 2/(r−2). Since (2.1) holds, the statement follows now directly from Theorem 2.5.
Powers of tight Hamilton cycles
Consider a tight Hamilton cycleCn(r,r−1)withnvertices which are ordered cyclically. Given an integer k, we define the k-th power Cn(r)(k)ofCn(r,r−1)to consist of all r-tuplesesuch that the maximum distance in this cyclic ordering between any two vertices ineis at mostr+k−2. Recall that in the graph case, the threshold for the appearance ofCn(2)(k)is known to be n−1/k fork ≥ 3[84]. If we count the edges ofCn(r)(k)by their leftmost vertex we gete(Cn(r)(k)) =n r+k−2r−1
forn≥r+ 2k−2.
Corollary 3.12. Let r ≥ 3 and k ≥ 2 be integers. Suppose thatp = ω(n−1/(r+i−2r−1 )), then the random hypergraphH(r)(n, p)contains a.a.s. a copy ofCn(r)(k). This threshold is asymptotically optimal.
Proof. One can argue similarly to Proposition 8.2 in [84] to showγ(Cn(r)(k))≤ r+k−2r−1
+Or,k(1/n).
The statement follows from Theorem 2.5. We omit the details.
27Since we can add each time one edge to connect two components and this doesn’t create any further cycles.
Chapter 4
Finding tight Hamilton cycles in random hypergraphs
In this chapter we prove28 Theorem 2.6, which resulted from collaboration with Allen, Koch, and Person [6]. We first give a short overview of the algorithm, then in Section 4.2 we prove the theorem, where the proofs of the Connecting and Reservoir Lemma are given in Section 4.3 and 4.4 respectively.
4.1 An informal algorithm overview
We briefly give some additional notation. Ans-tuple(u1, . . . , us)of vertices is an ordered set of distinct vertices. We often denote tuples by bold symbols, and occasionally also omit the brackets and write u = u1, . . . , us. Additionally, we may also use a tuple as a set and write for example, ifS is a set, S∪u:=S∪ {ui:i∈[s]}. Thereverseof thes-tupleuis thes-tuple←u−:= (us, . . . , u1).
In anr-uniform hypergraphGthe tupleP= (u1, . . . , u`)forms atight pathif the set{ui+1, . . . , ui+r} is an edge for every0≤i≤`−r. For anys∈[`]we say thatPstartswith thes-tuple(u1, . . . , us) =:v andendswith thes-tuple(u`−(s−1), . . . , u`) =: w. We also callvthestarts-tupleofP,wtheends -tupleofP, and P a v−w path. TheinteriorofP is formed by all its vertices but its start and end (r−1)-tuples. Note that the interior ofPis not empty if and only if` >2(r−1).
Overview of the algorithm
We start with the given sample of the random hypergraphH(r)(n, p)and we will reveal the edges as we proceed. First, using the Reservoir Lemma (Lemma 4.1 below), we construct a tight pathPres which covers a small but bounded away from zero fraction of[n], which has thereservoir property, namely that there is a setR⊂V(Pres)of size2Cp−1logn≤2n/log2nsuch that for anyR0⊂R, there is a tight path covering exactly the verticesV(Pres)\R0whose ends are the same as those ofPres, and this tight path can be found givenPresandR0in time polynomial inna.a.s.
We now greedily extendPres, choosing new vertices when possible and otherwise vertices inR. We claim that a.a.s. this strategy produces a structurePalmostwhich is almost a tight path extending Presand covering[n]. The reason it is onlyalmosta tight path is that some vertices inRmay be used twice. We denote the set of vertices used twice byR01. But we will succeed in covering[n]with high probability. Recall that, due to the reservoir property, we can dispense with the vertices fromR01in the partPresof the almost tight Hamilton pathPalmost.
28The proof presented in this chapter is a close adaption from [6].