Connecting Lemma - Connecting Lemma - Local density properties of Andrásfai graphs and powers o

3.2 Connecting Lemma

3.2.4 Connecting Lemma

Lemma 39. If vbc, vxy P E and |Nvpb, cq XNvpx, yq| “ m, then there are at least Ωpm²n²q quadruples pw₀, b₁, c₁, w₁q such that bcw₀b₁c₁w₁xy is

• a walk in H and

• a squared walk in L_v.

b c b1 c1 x y

Figure 3.4: Quadruple pw₀, b₁, c₁, w₁q that fulfills the conditions of Lemma 39, where thelink graph of v is indicated in green and hyperedges of H in red.

Proof. For every w P N_vpb, cq X N_vpx, yq Lemma 38 states that there are at least Ωpn²q walks inG_vw fromc tox of length 3. Let

X_b₁_c₁ “ twPN_vpb, cq XN_vpx, yq: cb₁c₁x is a walk in G_vwu for b₁, c₁ PV. Thus

pb1,c1qPV²

|Xb1c1| ěΩpmn²q and therefore the Cauchy-Schwarz inequality yields that

pb1,c1qPV²

|X_b₁_c₁|² ěΩpm²n²q.

If b₁, c₁ P V andw₀, w₁ P X_b₁_c₁, then bcw₀b₁c₁w₁xy has the desired properties.

Proposition 40. There is an integer K, such that for all edges abc, xyz P E and vertices v P Npa, b, cq XNpx, y, zq there are for some k “ kpabc, xyzq ď K withk”1 pmod 3qat leastΩpn^kqmanypu₁, . . . , u_kq PV^k for whichabcu₁. . . u_kxyz is

• a walk in H

• a squared walk in Lv.

Proof. Recall that in Proposition 33 we found an integer ` and a functiont:V^p2qÑ r`s such that for all distinctr, sPV there are Ωpn^tpr,sq´1q walks of lengthtpr, sqfromrtosinG_v. By the box principle there exists an integertď` such that the set QĎNvpb, cq ˆNvpx, yq of all pairs pu, u¹q PNvpb, cq ˆNvpx, yq with tpu, u¹q “t satisfies

|Q| ě |N_vpb, cq| ¨ |N_vpx, yq|

(3.11)

ě n² 16`.

For each walk v₀v₁. . . v_t inG_v there are by Definition32 at leastpβn²q^t many p2tq-tuples pb₁, c₁, . . . , b_t, c_tqsuch that

(i) biciv PE for i“1, . . . , t,

(ii) v₀ PN_vpb₁, c₁q and v_t PN_vpb_t, c_tq,

(iii) v_i PN_vpb_i, c_iq XN_vpb_i`1, c_i`1q for i“1, . . . , t´1 .

v_t vt´1

v₁ v₀

b c b1 c1 b2 c2 bt ct x y

Figure 3.5: A p3t`1q-tuple pv₀, v₁, . . . , v_t, b₁, c₁, . . . , b_t, c_tq P V^3t`1 satisfying (i), (ii), (iii), and (iv), where the link graph of v is indicated in green and

hyperedges of H in red.

Consequently, there are at least n²

16` ¨Ωpn^t´1q ¨ pβn²q^t“Ωpn^3t`1q

p3t`1q-tuples pv₀, v₁, . . . , v_t, b₁, c₁, . . . , b_t, c_tq P V^3t`1 satisfying (i), (ii), (iii) as well as

(iv) v₀ PN_vpb, cq and v_t PN_vpx, yq.

On the other hand, we can also write the number of thesep3t`1q-tuples as ÿ

ávPΨ

Ψ“ tpb₁, c₁, . . . , b_t, c_tq PV^2t: b_ic_iv PE for i“1, . . . , tu and for fixed^áv “ pb₁, c₁, . . . , b_t, c_tq PΨ

• I₀p^ávq “N_vpb, cq XN_vpb₁, c₁q

• I_ip^ávq “N_vpb_i, c_iq XN_vpb_i`1, c_i`1qfor i“1, . . . , t´1

• Itp^ávq “Nvpbt, ctq XNvpx, yq. Altogether we have thereby shown that

ávPΨ

ávPΨ

Ωp|I₀p^ávq|²n²q ¨. . .¨Ωp|I_tp^ávq|²n²q ěΩpn^2t`2qÿ

ávPΨ

p|I₀p^ávq| ¨. . .¨ |I_tp^ávq|q²

ěΩpn^2t`2q

´ ř

ávPΨ

|I₀p^ávq| ¨. . .¨ |I_tp^ávq|

¯2

|Ψ|

ěΩpn^2t`2q

´Ωpn^3t`1q n^t

¯2

“Ωpn^6t`4q p6t`4q-tuples, which fulfill the conditions of Proposition 40.

Since 6t`4”1 pmod 3qthis concludes the proof.

Definition 41. We call a sequence of vertices v₁. . . v_h a squared v-walk from abc to xyz with h interior vertices if abcv₁. . . v_hxyz is a walk in H and a squared walk in L_v.

Proposition 42. For all abc, xyz P E and v P Npa, b, cq XNpx, y, zq there are for some k¹ “k¹pabc, xyz, vq ď K `2 with k¹ ” 0 pmod 3q at least Ωpn^k¹q many squared v-walks with k¹ interior vertices from abc to xyz.

Proof. We choose vertices dPNvpb, cqand ePNvpc, dq, and with Proposition40 we find at least Ωpn^kq many squared v-walks from cde to xyz, where k “kpcde, xyzq ďK and k ”1 pmod 3q. Notice that if u₁. . . u_k is such a walk, then deu₁. . . u_k is a squared v-walk from abc toxyz.

Since |Nvpb, cq|,|Nvpc, dq| ě n{4 holds by (3.11), there are for some k ď K with k”1 pmod 3q at least ⁿ²_K^{16 “Ωpn²q pairs pd, eq with kpcde, xyzq “ k. Now altogether there are Ωpn^k`2q squared v-walks from abc to xyz with k`2 interior vertices. This implies Proposition42, sincek`2”0 pmod 3q.

Lemma 43. If abc, xyz P E and |Npa, b, cq XNpx, y, zq| “ m, then there is an integer t “ tpabc, xyzq ď pK`2q{3 such that at least Ωpm^t`1n^3tq squared walks from abc to xyz with 4t`1 interior vertices exist.

Proof. For every w P Npa, b, cq XNpx, y, zq Proposition 42 states that for some integer k¹ “ k¹pwq ď K ` 2 with k¹ ” 0 pmod 3q there are at least Ωpn^k¹q many squared w-walks from abc to xyz with k¹ interior vertices. By the box principle there exists an integer k² ď K `2 with k² ” 0 pmod 3q such that the set Q Ď Npa, b, cq X Npx, y, zq of all vertices w¹ P Npa, b, cq X Npx, y, zq with k¹pw¹q “ k² satisfies

|Q| ě |Npa, b, cq XNpx, y, zq|

K `2 “ m

K`2.

ForP “ pu₁, . . . , u_k²q PV^k² let X_P ĎQbe the set of verticesuP Qsuch thatP is a squared u-walk from abc to xyz. Since|Q| ěm{pK`2q, the average size of X_P is at least Ωpm{pK`2qq “Ωpmq by Proposition 42and double counting. Since

PPV^k²X_P^k²^{3`1 n^k² ě

´ř

PPV^k²XP

n^k²

¯k²{3`1

ěΩpm^k²^{3`1q, we get

PPV^k²

X_P^k²^{3`1 ěΩpm^k²^{3`1n^k²q.

Sincek² ”0 pmod 3qand every orderedk²-tupleP of vertices gives rise toX_P^k²^{3`1 squared walks fromabctoxyzwith 4k²{3`1 interior vertices, this implies Lemma43 with t “k²{3.

Finally we come to the main result of this section stated earlier as Proposition22.

Proposition 44 (Connecting Lemma). There are an integer M and ϑ_˚ ą 0, such that for all disjoint triples pa, b, cq and px, y, zq with abc, xyz P E there exists m ă M for which there are at least ϑ_˚n^m squared paths from abc to xyz with m internal vertices.

Proof. Recall that in Proposition 29 we found an integer ` and a functiont:V^p2qÑ r`s such that for all distinctr, sPV there are Ωpn^tpr,sq´1q walks of lengthtpr, sqfromrtosin G₃. By the box principle there exists an integertď` such that the setQĎNpa, b, cq ˆNpx, y, zqof pairs pu, u¹q PNpa, b, cq ˆNpx, y, zq with tpu, u¹q “t satisfies

|Q| ě |Npa, b, cq| ¨ |Npx, y, zq|

` ě n²

16`.

For each path v₀v₁. . . v_t inG₃ there are by Definition28 at leastpβn³q^t many p3tq-tuples pa₁, b₁, c₁, . . . , a_t, b_t, c_tq such that

(i) a_ib_ic_i P E fori“1, . . . , t

(ii) v₀ PNpa₁, b₁, c₁qand vtP Npat, bt, ctq

(iii) v_i PNpa_i, b_i, c_iq XNpa_i`1, b_i`1, c_i`1qfor i“1, . . . , t´1 . Consequently, there are at least

n²

16` ¨Ωpn^t´1q ¨ pβn³q^t“Ωpn^4t`1q

p4t`1q-tuples pv₀, . . . , v_t, a₁, b₁, c₁, . . . , a_t, b_t, c_tq P V^4t`1 satisfying (i), (ii), (iii) as well as

(iv) v₀ PNpa, b, cqand v_tPNpx, y, zq.

On the other hand, we can also write the number of thesep4t`1q-tuples as ÿ

ávPΨ

Ψ“ tpa1, b1, c1, . . . , at, bt, ctq PV^3t:aibici PE for i“1, . . . , tu and for fixed^áv “ pa₁, b₁, c₁, . . . , a_t, b_t, c_tq PΨ

v_t´1 v1

a b

c a1

c1 a2

c2 at

ct x y

Figure 3.6: A p4t ` 1q-tuple pv₀, . . . , v_t, a₁, b₁, c₁, . . . , a_t, b_t, c_tq P V^4t`1 satisfy-ing (i), (ii), (iii), and (iv), where orange quadruples indicate a copy of K₄^p3q, hyperedges of H are indicated in red, and green pairs are in the link graphof the corresponding v_i.

• I₀p^ávq “Npa, b, cq XNpa₁, b₁, c₁q

• I_ip^ávq “Npa_i, b_i, c_iq XNpa_i`1, b_i`1, c_i`1q for i“1, . . . , t´1

• Itp^ávq “Npat, bt, ctq XNpx, y, zq Altogether we have thereby shown that

ávPΨ

• t₀p^ávq “ tpabc, a₁b₁c₁q

• t_ip^ávq “ tpa_ib_ic_i, a_i`1b_i`1c_i`1q for i“1,2, . . . , t´1

• and t_tp^ávq “tpa_tb_tc_t, xyzq.

By the box principle there are Ψ_‹ Ď Ψ and a pt ` 1q-tuple pt₀, . . . , t_tq in r1,pK`2q{3s^t`1 such that

ávPΨ‹

i“0ti`1. Due to Lemma 43there are at least

ávPΨ‹

Ωp|I₀p^ávq|^t⁰^`1n^3t⁰q ¨. . .¨Ωp|I_tp^ávq|^t^t^`1n^3t^tq

“Ωpn³^ř^t^i“0^tⁱq ÿ

ávPΨ_‹

|I0p^ávq|^t⁰^`1¨. . .¨ |Itp^ávq|^t^t^`1

m-tuples, which up to repeated vertices fulfill the conditions of Proposition 44.

Let T “maxpt₀, . . . , t_tq. Since

n^T^pt`1q´^ř^t^i“0^tⁱ ÿ

ávPΨ‹

i“0

|I_ip^ávq|^tⁱ^`1 “ ÿ

ávPΨ‹

i“0

n^T^´tⁱ|I_ip^ávq|^tⁱ^`1

ě ÿ

ávPΨ‹

|I₀p^ávq|^T^`1¨. . .¨ |I_tp^ávq|^T^`1 “ ÿ

ávPΨ‹

p|I₀p^ávq| ¨. . .¨ |I_tp^ávq|q^T^`1

˜ ř

ávPΨ‹

|I₀p^ávq| ¨. . .¨ |I_tp^ávq|

|Ψ_‹|

¸T`1

¨ |Ψ_‹|

(3.13)

´Ωpn^4t`1q n^3t

¯T`1

¨n^3t ěΩpn^3t`pt`1qpT^`1qq,

which implies that

Ωpn³^ř^t^i“0^tⁱq ÿ

ávPΨ‹

|I₀p^ávq|^t⁰^`1¨. . .¨ |I_tp^ávq|^t^t^`1 ěΩpn^3t`pt`1q`^ř^t^i“0^tⁱ^`3^ř^t^i“0^tⁱq “Ωpn^mq.

At most Opn^m´1q tuples can fail being paths due to repeated vertices, thus there are Ωpn^mq squared paths from abc to xyz. This proves Proposition 44 with M “r4``4p``1q ¨ ^K`2₃ `2s, since

m“4t`4

i“0

t_i`1ď4``4p``1q ¨ K`2 3 `1.

Im Dokument Local density properties of Andrásfai graphs and powers of Hamiltonian cycles in hypergraphs (Seite 46-53)