On k-connectivity and Minimum Vertex Degree in Random s-intersection Graphs

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1 O k-coectivity ad Miimum Vertex Degree i Radom -Iterectio Graph Ju Zhao Oma Yağa Virgil Gligor CyLab ad Dept. of ECE Caregie Mello Uiverity {juzhao, oyaga, virgil}@adrew.cmu.edu Abtract Radom -iterectio graph have recetly received much iteret 9. I a radom -iterectio graph, each vertex i aiged to a et of item i ome maer, ad two vertice have a edge i betwee if ad oly if they hare at leat item. I particular, i a uiform radom -iterectio graph, each vertex idepedetly elect the ame umber of item uiformly at radom from a commo item pool, while i a biomial radom -iterectio graph, each item i ome item pool i idepedetly attached to each vertex with the ame probability. Thee two graph model have umerou applicatio; e.g., uig uiform radom -iterectio graph for cryptaalyi 4,5, ad to model ecure wirele eor etwork 8 0 ad olie ocial etwork, 2, ad uig a biomial radom -iterectio graph for cluterig aalyi 7, claificatio 8 ad the deig of itegrated circuit 34. For biomial/uiform radom -iterectio graph, we preet reult related to k-coectivity ad miimum vertex degree. Specifically, we derive the aymptotically exact probabilitie ad aymptotic zero oe law for the followig three propertie: (i k-vertex-coectivity, (ii k-edge-coectivity ad (iii the property of miimum vertex degree beig at leat k. Recetly, imilar reult have bee obtaied by Blozeli ad Rybarczyk 5, but their fidig are oly for uiform radom -iterectio graph, ot for biomial radom -iterectio graph, ad require parameter coditio which are dijoit from our our parameter coditio are ueful i practical eor etwork applicatio of the graph while their are ot. I term of biomial -iterectio graph, for the three propertie above, our paper report the firt reult o the exact probabilitie a well a the firt reult o the zero oe law. Keyword Radom iterectio graph, vertex coectivity, edge coectivity, vertex degree. Itroductio Radom -iterectio graph have received coiderable attetio recetly 9. I uch a graph, each vertex i equipped with a et of item i ome maer, ad two vertice etablih a udirected edge i betwee if ad oly if they have at leat item i commo. A large umber of work, 2, 4 9, tudy the cae of beig, uder which the graph are imply referred to a radom iterectio graph. Radom (-iterectio graph have bee ued to model ecure wirele eor etwork 8, 9 23, wirele frequecy hoppig 20, epidemic i huma populatio, 4, ocial etwork 2 4, 3 uch a collaboratio etwork 2 4 ad commo-iteret etwork, 2. Radom iterectio graph alo motivated Beer et al. 37, 38 to itroduce a geeral cocept of vertex radom graph that ubume ay graph model where radom feature are aiged to vertice, ad edge are draw baed o determiitic relatio betwee the feature of the vertice. Amog differet model of radom -iterectio graph, two widely tudied model are the o-called uiform radom -iterectio graph ad biomial radom -iterectio graph, which are defied i detail below.

2 . Graph model Uiform radom -iterectio graph. A uiform radom -iterectio graph, deoted by G (, K, P, i defied o vertice a follow. Each vertex idepedetly elect K differet item uiformly at radom from a pool of P ditict item. Two vertice have a edge i betwee if ad oly if they hare at leat item. The otio uiform mea that all vertice have the ame umber of item (but likely differet et of item. K ad P are both fuctio of, while doe ot cale with. It hold that K P. Biomial radom -iterectio graph. A biomial radom -iterectio graph, deoted by H (, t, P, i defied o vertice a follow. Each item from a pool of P ditict item i aiged to each vertex idepedetly with probability t. Two vertice etablih a edge i betwee if ad oly if they have at leat item i commo. The term biomial i ued ice the umber of item aiged to each vertex follow a biomial ditributio with parameter P (the umber of trial ad t (the ucce probability i each trial. t ad P are both fuctio of, while doe ot cale with. Alo it hold that P..2 Problem tatemet Our mai goal i thi paper i to ivetigate propertie related to k-coectivity ad miimum vertex degree of radom -iterectio graph (k-vertex coectivity ad k-edge coectivity are called together a k-coectivity for coveiece. I particular, we wih to awer the followig quetio: For uiform radom -iterectio graph G (, K, P (rep., biomial radom -iterectio graph H (, t, P, with parameter K (rep., t ad P calig with the umber of vertice, what i the aymptotic behavior of the probabilitie that G (, K, P (rep., H (, t, P i (i k-vertex-coected, (ii k-edge-coected, ad (iii ha a miimum vertex degree at leat k, repectively, a grow large? A graph i aid to be k-vertex-coected if the remaiig graph i till coected depite the deletio of at mot (k arbitrary vertice, ad k-edge-coectivity i defied imilarly for the deletio of edge 39; with k =, thee defiitio reduce to the tadard otio of graph coectivity 40. The degree of a vertex i defied a the umber of edge icidet o it. The three graph propertie coidered here are related to each other i that k-vertex-coectivity implie k-edge-coectivity, which i tur implie that the miimum vertex degree i at leat k Summary of Reult We ummarize our reult below, firt for a uiform radom -iterectio graph ad the for a biomial radom -iterectio graph. Throughout the paper, both ad k are poitive iteger ad do ot cale with. Alo, aturally we coider K P for graph G (, K, P ad P for graph H (, t, P. We ue the tadard Ladau aymptotic otatio Ω(, ω(, O(, o(, Θ(. PE deote the probability that evet E happe. k-coectivity ad miimum vertex degree i a uiform radom -iterectio graph. For uiform radom -iterectio graph G (, K, P uder P = Ω(, with equece α defied by! K 2 P = l + (k l l + α, if lim α = α,, the the followig covergece reult hold: 2

3 (i lim P G (, K, P i k-vertex-coected. = e e α (k!, (ii lim P G (, K, P i k-edge-coected. = e e α (k!, ad (iii lim P G (, K, P ha a miimum vertex degree at leat k. = e e α (k!. k-coectivity ad miimum vertex degree i a biomial radom -iterectio graph. For biomial radom -iterectio graph H (, t, P uder P = Ω( for 2 or P = Ω( c for = with ome cotat c >, with equece β defied by! t 2 P = l + (k l l + β, if lim β = β,, the the followig covergece reult hold: (i lim P H (, t, P i k-vertex-coected. = e e β (k!, (ii lim P H (, t, P i k-edge-coected. = e e β (k!, ad (iii lim P H (, t, P ha a miimum vertex degree at leat k. = e e β.4 Compario with related work (k!. Table o the ext page ummarize relevat work i the literature o uiform/biomial radom -iterectio graph i term of k-vertex-coectivity, k-edge coectivity, ad the property of miimum vertex degree beig at leat k. A oted i Footote, a zero oe law 23 mea that the probability that the graph ha certai property aymptotically coverge to 0 uder ome coditio ad coverge to uder ome other coditio. Amog the related work, Blozeli ad Rybarczyk 5 recetly derived the aymptotically exact probabilitie of uiform radom -iterectio graph for the three propertie above. Yet, their reult applie oly uder the coditio of log P = /; pecifically, their reult hold oly o the rage c / (l / P c 2 / (l 2/5 /, where c ad c 2 are ome poitive cotat. However, i the ecure wirele eor etwork applicatio of uiform radom -iterectio graph, P i expected to be at leat o the order of to eure that the etwork have reaoable reiliecy agait eor capture attack 9,2 23; amely, to obtai reult that are ueful i practice, it hold that P = Ω(. Clearly, becaue of c 2 / (l 2/5 / = o(, the rage of P aumed by Blozeli ad Rybarczyk 5 doe ot cover the practical rage of P = Ω(. The reult reported i thi paper cover the practical rage P = Ω( ad fill thi gap i the literature. I additio to extedig the literature o uiform radom -iterectio graph, a how i Table, we alo preet ovel reult for biomial radom -iterectio graph, icludig aymptotically exact probabilitie for k-coectivity ad the property of miimum vertex degree beig at leat k, uder (i geeral, ad k beig geeral or, or (ii beig, ad k beig geeral..5 Roadmap We orgaize the ret of the paper a follow. We detail the mai reult a theorem i Sectio 2. Sectio 3 ad 4 detail the tep of etablihig the theorem. We coclude the paper i Sectio 5. The Appedix provide additioal argumet ued i provig the theorem. The term e e α (k! equal 0 for α = ad for α =. Therefore, thi aymptotically exact probability reult alo implie a zero-oe law for the evet of iteret; i.e., a, the probability that the evet of iteret occur approache to if α ad to 0 if α. 3

4 Graph Property Reult Work k-coectivity & thi paper exact probability G (, K, P mi. vertex degree k 5(oly for log uiform radom P = / coectivity & 6(oly for -iterectio graph exact probability mi. vertex degree log P = / k-coectivity & exact probability 6 G (, K, P mi. vertex degree k zero-oe law, 26 coectivity & exact probability 25 mi. vertex degree zero-oe law 23, 24 k-coectivity & exact probability H (, t, P mi. vertex degree k zero-oe law coectivity & exact probability thi paper biomial radom mi. vertex degree zero-oe law -iterectio graph k-coectivity & exact probability H (, t, P mi. vertex degree k zero-oe law 26 coectivity & exact probability 27 mi. vertex degree zero-oe law 26, 28 Table : Compario of our reult with related work. k-vertex coectivity ad k-edge coectivity are together writte a k-coectivity. 2 Mai Reult Below we explai the mai reult of uiform radom -iterectio graph ad biomial radom -iterectio graph, repectively. 2. Reult of uiform radom -iterectio graph The followig theorem preet reult o k-coectivity ad miimum vertex degree i uiform radom -iterectio graph G (, K, P. Theorem For uiform radom -iterectio graph G (, K, P uder with equece α defied by the! K 2 P P = Ω(, ( = l + (k l l + α, (2 lim P G (, K, P i k-vertex-coected. = lim P G (, K, P i k-edge-coected. = lim P G (, K, P ha a miimum vertex degree at leat k. e e α (k!, if lim α = α (,, = 0, if lim α =,, if lim α =. 4 (3a (3b (3c

5 For the followig three propertie i graph G (, K, P : (i k-vertex coectivity, (ii k- edge-coectivity ad (iii the property of miimum vertex degree beig at leat k, we obtai aymptotically exact probabilitie from (3a ad zero-oe law from (3b ad (3c. A oted i Fooote o Page 3, with e e α (k! beig 0 for α = ad for α =, the right had ide of (3a, (3b, ad (3c ca together be writte a e e α (k! with lim α = α,. By Lemma 8 o Page 3, uder ( ad (2 with cotraied α = O(l l, we ca how that the left had ide of (2, i.e.,! K2 P i aymptotically equivalet to the edge probability of graph G (, K, P. A give i Lemma 3 o Page 4, with q deotig the edge probability of graph l +(k l l +α G (, K, P, if coditio (2 i replaced by q =, ad coditio ( i kept uchaged, the all reult i Theorem till follow. Hece, the uiform radom -iterectio graph model uder coditio ( exhibit the ame behavior with the well-kow Erdő-Réyi graph model 39 4, 2 i the ee that for each of (i k-vertex-coectivity, (ii k-edge-coectivity ad (iii the property of miimum vertex degree beig at leat k, a commo poit for the phae traitio from a zero-law to a oe-law occur whe the edge probability equal 2.2 Reult for biomial radom -iterectio graph l +(k l l. The followig theorem preet reult o k-coectivity ad miimum vertex degree i biomial radom -iterectio graph H (, t, P. Theorem 2 For biomial radom -iterectio graph H (, t, P uder { P = Ω(, for 2, P = Ω( c for ome cotat c >, for =, (4 with equece β defied by the! t 2 P = l + (k l l + β, (5 lim P H (, t, P i k-vertex-coected. = lim P H (, t, P i k-edge-coected. = lim P H (, t, P ha a miimum vertex degree at leat k. e e β (k!, if lim β = β (,, = 0, if lim β =,, if lim β =. For the three followig propertie i graph H (, t, P : (i k-vertex coectivity, (ii k-edgecoectivity ad (iii the property of miimum vertex degree beig at leat k, we obtai aymptotically exact probabilitie from (6a ad zero-oe law from (6b ad (6c. 2 I a Erdő-Réyi graph 39 4, a edge betwee each pair of vertice exit idepedetly with the ame probability. (6a (6b (6c 5

6 With e e β (k! beig 0 for β = ad for β =, the right had ide of (6a (6b ad (6c ca together be writte a e e β (k! with lim β = β,. A preeted i Lemma 2 o Page 4, uder (4 ad (5 with cotraied β = O(l l, we ca how that the left had ide of (5, i.e.,! t 2 P i aymptotically equivalet to the edge probability of graph H (, t, P. A give i Lemma 4 o Page 4, with ρ deotig the l +(k l l +β edge probability of graph H (, t, P, if coditio (5 i replaced by ρ =, ad coditio (4 i kept uchaged, the all reult i Theorem 2 till follow. Therefore, the biomial radom -iterectio graph model uder coditio (4 exhibit the ame behavior with Erdő- Réyi graph model, i the ee that for each of (i k-vertex-coectivity, (ii k-edge-coectivity, ad (iii the property of miimum vertex degree beig at leat k, a commo poit for the phae l +(k l l traitio from a zero-law to a oe-law occur whe the edge probability equal. The coditio (4 ha P = Ω( for 2, ad require a troger oe for = : P = Ω( c for ome cotat c >. The rage P = Θ( i covered by P = Ω(, but ot by P = Ω( c with c >. For = ad P = Θ(, reult for k-vertex-coectivity, k-edge coectivity, ad the property of miimum vertex degree beig at leat k ue a calig differet from (5, a give by 26, Theorem 4. 3 Etablihig Theorem Theorem i the pecial cae of = i proved by u 6. Below we explai the tep of etablihig Theorem for 2. I Sectio 3., we how that α ca be cofied a O(l l i provig Theorem. I Sectio 3.2, we coider the relatiohip betwee vertex coectivity, edge coectivity, ad miimum vertex degree. For uiform radom -iterectio graph G (, K, P, Table 2 ummarize ome otatio, which will be ued i the ret of the paper. Symbol Meaig Symbol Meaig κ v vertex coectivity V the et of vertice: {v, v 2,..., v } κ e edge coectivity S i the umber of item o vertex v i δ miimum vertex degree E ij the evet for the exitece of a edge betwee vertice v i ad v j q edge probability r mi ( P K, 2 Table 2: Notatio for uiform radom -iterectio graph G (, K, P. 3. Cofiig α To cofie α a O(l l i provig Theorem, we will demotrate Theorem uder α = O(l l Theorem. (7 Note that k-vertex-coectivity, k-edge-coectivity, ad the property of miimum vertex degree beig at leat k are all mootoe icreaig 3. For ay mootoe icreaig property I, the probability that a paig ubgraph (rep., upergraph of graph G ha I i at mot (rep., at leat the probability of G havig I. Therefore, to how (7, it uffice to prove the followig lemma. 3 A graph property i called mootoe icreaig if it hold uder the additio of edge 35, 36. 6

7 Lemma (a For graph G (, K, P uder P = Ω( ad! K 2 P = l + (k l l + α with lim α =, there exit graph G (, K, P uder P = Ω( ad! K 2 P = l + (k l l + α (8 (9 with lim α = ad α = O(l l, uch that there exit a graph couplig 4 uder which G (, K, P i a paig ubgraph of G (, K, P. (b For graph G (, K, P uder P = Ω( ad! K 2 P = l + (k l l + α with lim α =, there exit graph G (, K, P uder P = Ω( ad! K 2 P = l + (k l l + α with lim α = ad α = O(l l, uch that there exit a graph couplig uder which G (, K, P i a paig upergraph of G (, K, P. The proof of Lemma i provided i Sectio 6.2 i the Appedix. 3.2 Relatiohip betwee vertex coectivity, edge coectivity, ad miimum vertex degree Recall that the vertex coectivity of a graph i defied a the miimum umber of vertice that eed to be deleted to have the remaiig graph dicoected, ad the edge coectivity i defied imilarly for the deletio of edge 39. A how i Table 2 o Page 6, for graph G (, K, P, we ue κ v, κ e ad δ to deote the vertex coectivity, the edge coectivity, ad the miimum vertex degree, repectively. The k-vertex-coectivity, k-edge-coectivity, ad the property of miimum vertex degree beig at leat k, are give by evet κ v k, κ e k, ad δ k, repectively. For ay graph, the vertex coectivity i at mot the edge coectivity, ad the edge coectivity i at mot miimum vertex degree 39. Therefore, κ v κ e δ hold. The ad (0 ( P κ v k P κ e k P δ k, (2 k Pκ v k = Pδ k P (κ v < k (δ k P δ k P (κ v = l (δ > l. (3 Therefore, the proof i completed oce we how Lemma 2 ad 3 below. Note that ice k i a cotat, coditio (2 with α = O(l l i Theorem implie coditio (5 i Lemma 3. 4 A ued by Rybarczyk 26, 27, a couplig of two radom graph G ad G 2 mea a probability pace o which radom graph G ad G 2 are defied uch that G ad G 2 have the ame ditributio a G ad G 2, repectively. If G i a paig ubgraph (rep., upergraph of G 2, we ay that uder the couplig, G i a paig ubgraph (rep., upergraph of G 2, which yield that for ay mootoe icreaig property I, the probability of G havig I i at mot (rep., at leat the probability of G 2 havig I. l=0 7

8 Lemma 2 (7, Theorem 2 (our work For uiform radom -iterectio graph G (, K, P uder P = Ω(, if there exit equece α atifyig α = O(l l uch that! K 2 P = l + (k l l + α, the with δ deotig the miimum vertex degree, it hold that lim e α Pδ k = e (k!, if lim α = α,. (4 Lemma 3 For uiform radom -iterectio graph G (, K, P uder P = Ω( ad 2! K = P l ± O(l l, (5 the with κ v deotig the vertex coectivity ad δ deotig the miimum vertex degree, it hold for cotat iteger l that P (κ v = l (δ > l = o(. (6 We detail the proof of Lemma 3 below. 3.3 The proof of Lemma 3 A i Table 2 o Page 6, for graph G (, K, P, the et of vertice i V = {v, v 2,..., v }. Alo, for i =, 2,...,, we let S i deote the et of item o vertex v i. We itroduce evet E(J i the followig maer: E(J = vi T S i J T, (7 T V, T 2. where J = J 2, J 3,..., J i a ( -dimeioal iteger valued array, with J i defied through { max{ ( + ε K, λ K i }, i = 2,..., r, J i = (8 µ P, i = r +,...,, for a arbitrary cotat 0 < ε < ad ome poitive cotat λ, µ i Lemma 4 below, where r = mi ( P K, 2 a give i Table 2 o Page 6. By a crude boudig argumet, we get P (κ v = l (δ > l P E(J + P (κ v = l (δ > l E(J. (9 Hece, a proof of Lemma 3 coit of etablihig the followig two lemma. Note that uder (2 with α = O(l l, we have! K2 l ±O(l l P = = o( ad K P = o(, eablig u to ue Lemma 4 ad 5. Lemma 4 (, Propoitio 3 (our work If P = Ω( ad K P = o(, the for a arbitrary cotat 0 < ε < ad ome elected poitive cotat λ, µ, it hold that P E(J = o(. (20 8

9 Lemma 5 For uiform radom -iterectio graph G (, K, P uder P = Ω( ad the 2! K = P l ± O(l l, (2 P (κ v = l (δ > l E(J = o(. (22 The proof of Lemma 5 i give i Sectio 6.3 i the Appedix. 4 Etablihig Theorem 2 Similar to the idea of cofiig α i Theorem, here we cofie β a O(l l i Theorem 2. Specifically, we will demotrate Theorem 2 uder β = O(l l Theorem 2. (23 Sice k-vertex-coectivity, k-edge-coectivity, ad the property of miimum vertex degree beig at leat k, are all mootoe icreaig, the to how (23, it uffice to prove the followig lemma. Lemma 6 (a For graph H (, t, P uder! t 2 P = l + (k l l + β (24 with lim β =, there exit graph H (, t, P uder! 2 l + (k l l + t β P = (25 with lim β = ad β = O(l l uch that there exit a graph couplig uder which H (, t, P i a paig ubgraph of H (, t, P. (b For graph H (, t, P uder! t 2 P = l + (k l l + β (26 with lim β =, there exit graph H (, t, P uder! 2 l + (k l l + t β P = (27 with lim β = ad β = O(l l uch that there exit a graph couplig uder which H (, t, P i a paig upergraph of H (, t, P. The proof of Lemma 6 i detailed i Sectio 6.4 i the Appedix. Now we ue Theorem to prove Theorem 2 with cofied β = O(l l. Here, the mai idea i to exploit a couplig reult betwee the uiform -iterectio graph ad a biomial -iterectio 9

10 graph. Let I deote either oe of the followig three graph propertie: k-vertex-coectivity, k- edge coectivity, ad the property of miimum vertex degree beig at leat k. With K ad K + defied by we have from Lemma that if t P = ω(l, the K ± = t P ± 3 l (l + t P, (28 P Graph G (, K, P ha I. o( P Graph H (, t, P ha I. P Graph G (, K +, P ha I. + o(. (29 Uder coditio (4, (5, ad β = O(l l, we ow how that t P = ω(l. From (5 ad β = O(l l, we firt get From (4 ad (30, it follow that! t 2 P l ± O(l l = = l ± o(. (30 t P = t 2 P P {{! l ± o( } 2 Ω( = Ω ( 2 2 (l 2, for 2, = {! l ± o( } 2 Ω( c = Ω ( (3 c 2 (l 2, for =, yieldig t P = ω(l i view of c >, o we ca ue (29. Uig (28 ad (3, we further obtai (K ± 2 t P ± 3 l (l + t P 2 = P P = (t P 2 ( 2 3 l l ± + P t P t P ( = t 2 P ± o, (32 l where i the lat tep we ue t P = ω ( (l 3, which follow from (3 due to cotat c >. Applyig (5 ad (30 to (32, we have! (K± 2 P I view of (33 ad P = Ω(, we ue Theorem to obtai = l + (k l l + β ± o(. (33 lim P Graph G (, K ±, P ha I. = e e lim β±o( (k! The proof of Theorem 2 i completed by (29 ad (34. 5 Cocluio e lim β = e (k!. (34 Radom -iterectio graph have bee ued i a wide rage of applicatio. Two widely tudied model are uiform radom -iterectio graph ad biomial radom -iterectio graph. I thi paper, for a uiform/biomial radom -iterectio graph, we derive exact aymptotic expreio for the probabilitie of the followig three propertie: (i the graph i k-vertex-coected, (ii the graph i k-edge-coected, ad (iii each vertex ha at leat k eighborig vertice. 0

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12 9 J. Zhao, O. Yağa, ad V. Gligor, Secure k-coectivity i wirele eor etwork uder a o/off chael model, i IEEE Iteratioal Sympoium o Iformatio Theory (ISIT, J. Zhao, O. Yağa, ad V. Gligor, Coectivity i ecure wirele eor etwork uder tramiio cotrait, to appear i Allerto Coferece o Commuicatio, Cotrol, ad Computig, 204. Available olie at 2 L. Echeauer ad V. Gligor, A key-maagemet cheme for ditributed eor etwork, i ACM Coferece o Computer ad Commuicatio Security (CCS, O. Yağa, Radom Graph Modelig of Key Ditributio Scheme i Wirele Seor Network. PhD thei, Dept. of ECE, Uiverity of Marylad, Jue 20. Available olie at 23 O. Yağa ad A. M. Makowki, Zero oe law for coectivity i radom key graph, IEEE Traactio o Iformatio Theory, vol. 58, pp , May S. Blackbur ad S. Gerke, Coectivity of the uiform radom iterectio graph, Dicrete Mathematic, vol. 309, o. 6, Augut K. Rybarczyk, Diameter, coectivity ad phae traitio of the uiform radom iterectio graph, Dicrete Mathematic, vol. 3, K. Rybarczyk, Sharp threhold fuctio for the radom iterectio graph via a couplig method, The Electroic Joural of Combiatoric, vol. 8, pp , K. Rybarczyk, The couplig method for ihomogeeou radom iterectio graph, ArXiv e-prit, Ja Available olie at 28 K. B. Siger-Cohe, Radom iterectio graph. PhD thei, Departmet of Mathematical Sciece, The Joh Hopki Uiverity, M. Bradojić, A. Hagberg, N. W. Hegarter, N. Lemo, ad A. G. Percu, The phae traitio i ihomogeeou radom iterectio graph, Arxiv e-prit, 203. Available olie at 30 J. Jaworki, M. Karońki, ad D. Stark, The degree of a typical vertex i geeralized radom iterectio graph model, Dicrete Mathematic, vol. 306, o. 8, pp , M. Blozeli ad J. Damaracka, Degree ditributio of a ihomogeeou radom iterectio graph., The Electroic Joural of Combiatoric, vol. 20, o. 3, p. P3, S. Nikoletea, C. Raptopoulo, ad P. Spiraki, Large idepedet et i geeral radom iterectio graph, Theororetical Computer Sciece, vol. 406, pp , Oct D. Stark, The vertex degree ditributio of radom iterectio graph, Radom Structure & Algorithm, vol. 24, o. 3, pp , M. Karońki, E. R. Scheierma, ad K. B. Siger-Cohe, O radom iterectio graph: The ubgraph problem, Combiatoric, Probability ad Computig, vol. 8, pp. 3 59, Ja S. Jao, T. Luczak, ad A. Rucińki, Radom graph. Wiley-Iterciece Serie o Dicrete Mathematic ad Optimizatio, B. Bollobá, Radom graph. Cambridge Studie i Advaced Mathematic, E. Beer, J. A. Fill, S. Jao, ad E. R. Scheierma, O vertex, edge, ad vertex-edge radom graph, The Electroic Joural of Combiatoric, vol. 8, o., p. P0, E. Beer, J. A. Fill, S. Jao, ad E. R. Scheierma, O vertex, edge, ad vertex-edge radom graph, i SIAM: Aalytic Algorithmic ad Combiatoric (ANALCO, P. Erdő ad A. Réyi, O the tregth of coectede of radom graph, Acta Mathematica Academiae Scietiarum Hugaricae, pp , 96. 2

13 40 P. Erdő ad A. Réyi, O radom graph, I, Publicatioe Mathematicae (Debrece, vol. 6, pp , E. N. Gilbert, Radom graph, The Aal of Mathematical Statitic, vol. 30, pp. 4 44, Appedix We firt preet i Sectio 6. additioal lemma ued i provig the theorem. Afterward, we detail the proof of the lemma. 6. Additioal lemma Some additioal lemma are give below. The relatio tad for aymptotically equivalece; i.e., f g mea lim (f /g =. Lemma 7 If! K2 l ±O(l l P = ad P = Ω( c for cotat c, the K = Ω ( c 2 2 (l 2. Lemma 8 The followig propertie (a ad (b hold, where q i the edge probability i uiform radom -iterectio graph G (, K, P, a oted i Table 2 o Page 6. = o ( (a If P = Ω( ad! K2 P = (b If P = Ω( ad q = l ±O(l l, the q! K2 P l ±O(l l, the q! K2 P ad q! K2 P = o (. ad q! K2 P Lemma 9 For uiform radom -iterectio graph G (, K, P uder K = ω(, the followig propertie (a (b ad (c hold for i = r +, r + 2,..., (i.e., vertex v i / {v, v 2,..., v r }, where a give i Table 2 o Page 6, E ij deote the evet that a edge exit betwee vertice v i ad v j, S i i the umber of item o vertex v i, ad q i the edge probability. (a If r j= S j ( + ε K for ome poitive cotat ε, the for ay poitive cotat ε 2 < ( + ε, it hold for all ufficietly large that r P j=. E ij S, S 2,..., S r e q(+ε 2. (35 (b If r j= S j λ rk for ome poitive cotat λ, the for ay poitive cotat λ 2 < λ, it hold for all ufficietly large that r P j= E ij S, S 2,..., S r e λ 2rq. (36 (c If r j= S j µ P for ome poitive cotat µ, the for ay poitive cotat µ 2 < (! µ, it hold for all ufficietly large that r P j= E ij S, S 2,..., S r e µ 2K. (37 Lemma 0 For uiform radom -iterectio graph G (, K, P uder P = Ω(, K = ω( ad r = ω( (ote that r = mi ( P K, 2 a give i Table 2 o Page 6, the followig propertie (a (b ad (c hold for ay cotat iteger R 2, where ε, λ ad µ are pecified i Lemma 4. 3

14 (a Let ε 3 be ay poitive cotat with ε 3 < ( + ε. For all ufficietly large, it hold for r = 2, 3,..., R that P A l,r E(J r r 2 r q (2rq l e q(+ε3. (b Let λ 2 be ay poitive cotat with λ 2 < λ. For all ufficietly large, it hold for r = R +, R + 2,..., r that P A l,r E(J r r 2 r q e λ 2rq /3. (c Let µ 2 be ay poitive cotat with µ 2 < (! µ. For all ufficietly large, it hold for r = r +, r + 2,..., l 2 that P A l,r E(J e µ 2K /3. Lemma (8, Lemma 4 Let K ad K + deote t P 3 l (l + t P ad t P + 3 l (l + t P, repectively. If t P = ω(l, the for ay mootoe icreaig graph property I, it hold that P Graph G (, K, P ha I. o( P Graph H (, t, P ha I. P Graph G (, K +, P ha I. + o(. Lemma 2 The followig propertie (a ad (b hold, where ρ i the edge probability i biomial radom -iterectio graph H (, t, P. l ±O(l l (a Uder (4 ad! t 2 P =, the ρ! t 2 P ad ρ! t 2 P = o (. l ±O(l l (b Uder (4 ad ρ =, the ρ! t 2 P ad ρ! t 2 P = o (. Lemma 3 Theorem till follow with (2 replaced by q = l + (k l l + α, (38 where q deote the edge probability of uiform radom -iterectio graph G (, K, P. I other word, for uiform radom -iterectio graph G (, K, P uder (, with equece α defied by (38, we have lim P G (, K, P i k-vertex-coected. = lim P G (, K, P i k-edge-coected. = lim P G (, K, P ha a miimum vertex degree at leat k. = e e α (k!, if lim α = α,. Lemma 4 Theorem 2 till follow with (5 replaced by ρ = l + (k l l + β, (39 4

15 where ρ deote the edge probability of biomial radom -iterectio graph H (, t, P. I other word, for biomial radom -iterectio graph H (, t, P uder (4, with equece β defied by (39, we have 6.2 Proof of Lemma lim P H (, t, P i k-vertex-coected. = lim P H (, t, P i k-edge-coected. = lim P H (, t, P ha a miimum vertex degree at leat k. = e e β (k!, if lim β = β,. Provig property (a. We defie α by α = max{α, l l }, (40 ad defie K uch that! ( K P 2 = l + (k l l + α. (4 We et K := K, (42 ad P := P. (43 From (0 (40 ad (4, it hold that K K. (44 The by (42 (44 ad the fact that K ad K are both iteger, it follow that K K. (45 From (43 ad (45, by 8, Lemma 3, there exit a graph couplig uder which G (, K, P i a paig ubgraph of G (, K, P. Therefore, the proof of property (a i completed oce we how α defied i (9 atifie We firt prove (46. From (9 (4 ad (42, it hold that which together with (40 ad lim α = yield (46. lim =, (46 α = O(l l. (47 α α, (48 5

16 Now we etablih (47. From (42, we have K > K. The from (9 ad (43, it hold that α =! K 2 l + (k l l P >! ( K 2 l + (k l l. (49 P By lim α =, it hold that α 0 for all ufficietly large. The from (40, it follow that α = O(l l, (50 which alog with Lemma 7, equatio (4 ad coditio P = Ω( iduce K = Ω ( (l 2. (5 Hece, we have lim K = ad it further hold for all ufficiet large that ( K 2 > ( K 2 3( K 2. (52 Applyig (52 to (49 ad the uig (4, Lemma 7 ad P = Ω(, it follow that α >! ( K 2 3( K 2 l + (k l l P = α 3! Θ( P (l 2 = α O ( (l 2. (53 A oted at the begiig of Sectio 3, our proof i for 2 ice the cae of = already i proved by u 6. Uig 2 i (53, it hold that α > α + o(, which alog with (48 ad (50 yield (47. Provig property (b. We defie α by α = mi{α, l l }, (54 ad defie K uch that! ( K P 2 = l + (k l l + α. (55 We et K := K, (56 ad P := P. (57 6

17 From (0 (54 ad (55, it hold that K K. (58 The by (56 (58 ad the fact that K ad K are both iteger, it follow that K K. (59 From (57 ad (59, by 8, Lemma 3, there exit a graph couplig uder which G (, K, P i a paig upergraph of G (, K, P. Therefore, the proof of property (b i completed oce we how α defied i ( atifie We firt prove (60. From ( (55 ad (56, it hold that lim =, (60 α = O(l l. (6 α α, (62 which together with (54 ad lim α = yield (60. Now we etablih (6. From (56, we have K < K +. The from ( ad (57, it hold that α =! K 2 l + (k l l P <! ( K 2 + l + (k l l. (63 P By lim α =, it hold that α 0 for all ufficietly large. The from (54, it follow that α = O(l l, (64 which alog with Lemma 7, equatio (55 ad coditio P = Ω( iduce K = Ω ( (l 2. (65 Hece, we have lim K = ad it further hold for all ufficiet large that ( K + 2 < ( K 2 + 3( K 2. (66 Applyig (66 to (63 ad the uig (55, Lemma 7 ad P = Ω(, it follow that α <! ( K 2 + 3( K 2 l + (k l l P = α + 3! Θ( P (l 2 = α + O ( (l 2. (67 A oted at the begiig of Sectio 3, our proof i for 2 ice the cae of = already i proved by Rybarczyk 25. Uig 2 i (67, it hold that α < α + o(, which alog with (62 ad (64 yield (6. 7

18 6.3 The proof of Lemma 5 By the aalyi i, Sectio IV, we obtai, Equatio (48. Namely, with ome evet defied a follow: C r : evet that the iduced ubgraph of G (, K, P defied o vertex et {v, v 2,..., v r } i coected, B l,r : evet that ay vertex i {v r+, v r+2,..., v r+l } ha a edge with at leat oe vertex i {v, v 2,..., v r }, D l,r : evet that ay vertex i {v r+l+, v r+l+2,..., v } ad ay vertex i {v, v 2,..., v r } ha o edge i betwee, ad A l,r : evet that evet C r, B l,r ad D l,r all happe, it hold that P (κ = l (δ > l E(J l 2 r=2 ( l ( l r P A l,r E(J. (68 ad The proof of Lemma 5 i completed oce we how the followig three reult: R ( ( l l r ( ( l l r r=2 r r=r+ P A l,r P A l,r E(J = o(, (69 E(J = o(, (70 l 2 r=r + ( ( l l r P A l,r E(J = o(, (7 where r = mi ( P K, 2 a give i Table 2 o Page 6. From coditio (2, it follow that K P = o(, yieldig r = ω(. From coditio (2 ad P = Ω(, we ue Lemma 7 to derive K = ω(. Therefore, we have P = Ω(, K = ω( ad r = ω(, eablig u to ue Lemma 0. I additio, give coditio (2 ad P = Ω(, we ue Lemma 8 to obtai Hece, it hold that q = l ± O(l l. (72 ad there exit cotat c 0 uch that q 2 l, for all ufficietly large, (73 q l c 0 l l, for all ufficietly large. (74 8

19 6.3. Etablihig (69 From ( l l, ( l r r ad property (a of Lemma 0, it follow that ( l ( l r P A l,r E(J l r r r 2 q r (2rq l e q(+ε 3 = (2r l r r 2 l+r q l+r e q(+ε 3. (75 Applyig (72 ad (74 to (75, we get ( ( l P A l,r E(J l r ( 2 l l+r (2r l r r 2 l+r e (+ε 3(l c 0 l l 2 l+r (2r l r r 2 ε 3 (l l+r +c 0(+ε 3 = o(. Sice R i a cotat, we the obtai R ( ( l l r r=2 P A l,r E(J = o( Etablihig (70 From ( l l, ( ( r l r e( l ( r e r r ad property (b of Lemma 0, we have ( ( l ( e( l r P A l,r E(J l r r 2 q r e λ 2rq /3 l r r l+r e r q r e λ 2rq /3. (76 Applyig (72 ad (74 to (76, we get ( ( l ( 2 l r P A l,r E(J l+r e r e λ 2r(l c 0 l l /3 l r l+ (2e λ 2/3 (l c 0λ 2 /3+ r. (77 Give 2e λ2/3 (l c 0λ 2 /3+ = o( ad (77, we obtai r ( ( l P A l,r E(J l+ (2e λ2/3 (l c 0λ 2 /3+ r l r r=r+ r=r+ ( 2e = l+ λ 2 /3 (l c 0λ 2 /3+ R+ 2e λ2/3 (l c 0λ 2 /3+ l+ λ 2(R+/3 ( 2e(l c 0λ 2 /3+ R+. (78 We pick cotat R 3(l+ λ 2 o that l + λ 2 (R + /3 λ 2 3. A a reult, we obtai ad thu r r=r+ R.H.S. of (78 = o( P A l,r E(J = o(. 9

20 6.3.3 Etablihig (7 From ( l l ad property (c of Lemma 0, it hold that l 2 r=r + ( ( l l r P A l,r E(J l e µ 2K /3 Give coditio P = Ω( ad (2, we ue Lemma 7 to derive which yield K = Ω ( 2 2 (l 2 = ω(, l 2 r=r + ( l. (79 r µ 2 K /3 2 l 2, for all ufficietly large. (80 We have l 2 r=r + ( l r l 2 r=r + ( r r=0 ( = 2. (8 r Applyig (80 ad (8 to (79, we fially obtai l 2 r=r + ( ( l l r P A l,r E(J l 2 e µ 2K /3 = e l l + l 2 µ 2K /3 e l l l 2, for all ufficietly large. (82 With, it i clear that l 2 r=r + ( ( l l r P A l,r E(J = o(. ( Proof of Lemma 6 (a We et P = P, (84 ad β = max{β, l l }. (85 Give (85 ad lim β =, we clearly obtai lim β = ad β = O(l l. It hold from (85 that β β, which alog with (24 (25 ad (84 yield t t. Uder t t ad P = P, by 26, Sectio 3, there exit a graph couplig uder which H (, t, P i a paig ubgraph of H (, t, P. (b We et P = P, (86 20

21 ad β = mi{β, l l }. (87 Give (87 ad lim β =, we clearly obtai lim β = ad β = O(l l. It hold from (87 that β β, which alog with (26 (27 ad (86 yield t t. Uder t t ad P = P, by 26, Sectio 3, there exit a graph couplig uder which H (, t, P i a paig upergraph of H (, t, P. 6.5 Proof of Lemma 7 From coditio 2! K = P l ± O(l l l, (88 it hold that K 2 P = Θ ( (l, (89 which alog with coditio P = Ω( c yield K = P Θ ( (l = Ω ( c 2 2 (l 2. ( Proof of Lemma 8 We prove propertie (a ad (b below, repectively. (a Here we demotrate property (a. We till have (88 ad (89 here. The ettig c a i (90, it hold that K = Ω ( 2 2 (l 2. (9 Give (89 ad (9, we ue 7, Lemma ad, Lemma 8 to have q = { K 2 (!( P ± O K 2 ( P ± O K, for 2, K 2 P ± O ( K 2 (92 P, for =. Now we ue (89 (9 ad (92 to derive q! K2 P Firt, (89 ad (9 imply K2 P = o ( ad q! K2 P. = o( ad K = ω(, repectively, which are ued i (92 to derive q! K2 P. Secod, applyig (89 ad (9 directly to (92, we obtai the followig two cae: (i For 2, it hold that q =! =! ( K 2 P ( 2 K P ( l + Θ ( ± o. ± O ( ( (l ± O 2 2 (l 2 2

22 (ii For =, it hold that q = K 2 P ( (l 2 ± O = K 2 P ( ± o. Summarizig cae (i ad (ii above, we have proved property (a of Lemma 8. (b Now we how property (b. By 2, Lemma 6, the edge probabilty q atifie q ( K 2 ( P. (93 From (93 ad coditio q = l ±O(l l, it hold that ( K 2 ( P l o(, which alog with ( K 2 ( P! K 2 (P + ad P = Ω( lead to Therefore, it follow that K 2! (P + l o( = Ω( l. K = Ω ( 2 2 (l 2. (94 Note that if for ome, it hold that P < 2K, the two vertice hare at leat item l ±O(l l with probability, reultig i q =. Therefore, give coditio q =, we kow that for all ufficietly large, P 2K hold, o the probability that two vertice hare exactly item i expreed by ( K ( P K /( P K K. The which together with coditio q = q P Two vertice hare exactly item. ( ( /( K P K P = K K = 2 K! i=0 (P K (K i K i=0 (P i i=0! (K + 2 P, (95 l ±O(l l implie (K + 2 = O ( (l. (96 P From (94 ad the fact that i a cotat, it hold that K + K, which with (96 yield K 2 P = O ( (l. (97 22

23 Now we ue (97 (94 ad (92 to derive q! K2 P ad q! ( K2 P = o, i a way imilar to provig property (a above. Firt, (97 ad (94 imply K = ω( ad K2 P = o(, repectively, which are ued i (92 to derive q! K2 P. Secod, applyig (97 ad (94 directly to (92, we till have cae (i ad (ii i the proof of property (a above. Hece, fially we alo obtai q! ( K2 P = o. The property (b i proved. 6.7 Proof of Lemma 9 Recall that E ij deote the evet that a edge exit betwee vertice v i ad v j, ad S i i the umber of item o vertex v i. Evet E ij occur if ad oly if S i S j. Therefore, evet r j= E ij i equivalet to r ( j= Si S j <, which clearly i implied by evet Si ( r j= S j <. The r P j= ( r E ij S, S 2,..., S r P S i j= S j < S, S 2,..., S r ( r = P S i S j S, S 2,..., S r j= ( r P S i S j = S, S 2,..., S r = = e ( j= ( r j= S ( j P K ( P K ( r j= S ( j K ( P rj= S j ( K ( P. (98 Firt, we have (93 by 2, Lemma 6. Applyig (93 to (98, we obtai r P j= E ij S, S 2,..., S r e ( rj= S j ( K q. (99 Now we prove propertie (a, (b, ad (c of Lemma 9, repectively. (a Give coditio K = ω(, it follow that ( + ε K > for all ufficietly large. For property (a, we have coditio r j= S j ( + ε K, which i ued i (99 to derive r ( (+εk P E ij S, S 2,..., S r e ( K q. (00 j= We have ( (+ε K ( K = i=0 { ( + ε K i } i=0 (K i ( + ε K. (0 K 23

24 Give coditio ε 2 < ( + ε ad K = ω(, it follow that K + +ε +ε 2 for all ufficietly large, yieldig ( + ε K K ( + ε ( + + ε + ε2 + = + ε 2. (02 Applyig (02 to (0, we obtai ( (+ε K ( K which i ubtituted ito (00 to iduce r P j= ( ( + ε2 = + ε2, E ij S, S 2,..., S r e q(+ε 2. (b Give coditio K = ω(, it follow that λ rk > for all ufficietly large. For property (b, we have coditio r j= S j λ rk, which i ued i (99 to derive We have ( λ rk r ( λrk P E ij S, S 2,..., S r e ( K q. (03 j= ( K = i=0 ( λ rk i i=0 (K i Give coditio λ 2 < λ ad K = ω(, it follow that K + ufficietly large, iducig Applyig (05 to (04, we obtai ( λ rk. (04 K λ + λ 2 r(λ λ 2 for all λ rk λ r ( + r(λ λ2 = λ 2 r. (05 K + ( λ rk ( K which i ubtituted ito (03 to iduce r P j= ( λ2 r = λ2 r λ 2 r, E ij S, S 2,..., S r e λ 2rq. (c From P K = ω(, it follow that P = ω(. The µ P > for all ufficietly large. For property (c, we have coditio r j= S j µ P, which i ued i (98 to derive r P j= E ij S, S 2,..., S r ( µp ( K e ( P. (06 24

25 We have ( µ P ( K ( P (! ( µ P (! (K (! (P! ( µ P (K. (07 P Give 0 < µ 2 < (! µ + ad P K = ω(, it follow that P µ 2 µ ad K!µ 2 for all ufficietly large, iducig +!µ2 µ µ P P µ ( + µ 2 µ!µ 2 + = µ 2!µ 2, (08 ad (K K K K (!µ 2!µ 2 = µ µ K. (09 Applyig (08 ad (09 to (07, we obtai ( µ P ( K ( P! ( µ which i ubtituted ito (06 to iduce 6.8 Proof of Lemma 0 By defiitio, we have ad r P j= 2!µ 2!µ 2 µ K = µ 2 K, E ij S, S 2,..., S r e µ 2K. B l,r := D l,r := r+l l+r i=r+ j=l+ i=l+r+ j= E ij, r E ij, A l,r := B l,r C r D l,r. The By, Lemma, P A l,r E(J = P C r B l,r D l,r E(J P C r r r 2 q r. (0 25

26 We have { r } P B l,r S, S 2,..., S r = P E ij S, S 2,..., S r l. ( j= By the uio boud, r r r P E ij S, S 2,..., S r PE ij S, S 2,..., S r = PE ij = rq. (2 j= j= j= The P B l,r S, S 2,..., S r (rq l. (3 We have { ( r P D l,r E(J S, S 2,..., S r = P E ij j= } l r. E(J S, S 2,..., S r (4 The by Lemma 9, for all ufficietly large, (a for r = 2, 3,..., R, it hold that P D l,r E(J S, S 2,..., S r e q(+ε 2( l r e q(+ε 3. (5 To ee thi, pick ay ε 3 < (+ε, ad ue Lemma 9 with ε 2 atifyig ε 3 < ε 2 < (+ε. (b for r = 2, 3,..., r, it hold that P D l,r E(J S, S 2,..., S r e λ 2rq ( l r e λ 2rq /3. (6 (c for r = r +, r + 2,..., l 2, it hold that 6.9 Proof of Lemma 2 Sice i a cotat, from coditio it hold that P D l,r E(J S, S 2,..., S r e µ 2K ( l r e µ 2K /3. (7! t 2 P l ± O(l l = = Θ t 2 P = Θ ( (l ( l, (8 = o(. (9 Note that uder (4, we have P = Ω( for 2 or P = Ω( c for = with ome cotat c >. Hece, uder (4 ad (9, we ue 8, Propoitio 2 ad obtai ( P 2 ρ = t ± O(t 2 P { =! t 2 P ± O(t 2 P ± O(P, for 2, t 2 P ± O(t 2 P (20, for =. 26

27 Furthermore, applyig P = Ω( ad (9 to (20, we derive ρ! t 2 P ad {! ρ = t 2 P + Θ ( ( l ± Θ (l ± O( c, for 2, t 2 P + O ( ( l 2, for =, = (! t 2 P + o. (2 6.0 Proof of Lemma 3 Lemma 3 i the pecial cae of = i proved by u 6. Below we explai the tep of etablihig Theorem for 2. I Sectio 3., we how that α ca be cofied a O(l l i provig Theorem. I Sectio 3.2, we coider the relatiohip betwee vertex coectivity, edge coectivity, ad miimum vertex degree. We coider the followig two cae: (a lim α = α (,, (b lim α =, ad (c lim α =. (a For cae (a with lim α = α (,, it i clear that α i bouded; i.e., α = O(. From q = ad α = O(, we ue Lemma 8 to obtai! K2 l +(k l l +α ±o( l +(k l l +α P =. The we apply Theorem ad clearly etablih Lemma 3 for thi cae (a, i view of lim α ± o( = lim α = α. (b We ow coider cae (b with lim α =. Firt, we have (93 by 2, Lemma 6. From (93, it hold that! K 2 (P + ( K 2 ( P q. (22 Give coditio P = Ω( ad the fact that i a cotat, we have ( P = (P + P + P ( = (P + P ( = (P + Θ. (23 From (22 ad (23, it follow that 2! K q P Θ which alog with lim α = yield that there exit ome γ with lim γ = uch that (,! K 2 P = l + (k l l + γ. The we apply Theorem ad clearly prove Lemma 3 for thi cae (b. 27

28 (c We ow coider cae (c with lim α =. Clearly, it hold that α < 0 for all ufficietly large. We firt how that the reult i true whe α i cofied a O(l l ad the demotrate that α ca be ideed cofied a O(l l. l +(k l l +α Firt, whe α i cofied a O(l l. From q = ad α = O(l l, l +(k l l +α±o( we ue Lemma 8 to obtai! K2 P =. The we apply Theorem ad the reult clearly follow i view of lim α ± o( = lim α =. Secod, to how α ca be ideed cofied a O(l l, imilar to Lemma, we how the followig: For graph G (, K, P uder P = Ω( ad edge probability q = l + (k l l + α (24 with lim α =, there exit graph G (, K, P uder P = Ω( ad edge probability q = l + (k l l + α (25 with lim α = ad α = O(l l, uch that there exit a graph couplig uder which G (, K, P i a paig ubgraph of G (, K, P. The proof of the above cofiig argumet i imilar to property (a of Lemma, a detailed below. We defie α by α = max{α, l l }, (26 ad defie K uch that the edge probability q of graph G (, K, P We et q = l + (k l l + α. (27 K := K, (28 ad P := P. (29 From q = l +(k l l +α, (26 ad (27, it i clear that K K. (30 The by (28 (30 ad the fact that K ad K are both iteger, it follow that K K. (3 From (29 ad (3, by 8, Lemma 3, there exit a graph couplig uder which G (, K, P i a paig ubgraph of G (, K, P. Therefore, the proof of property (a i completed oce we how α defied i (25 atifie lim =, (32 α = O(l l. (33 28

29 We firt prove (32. From (25 (27 ad (28, it hold that α α, (34 which together with (26 ad lim α = yield (32. Now we etablih (33. Firt, by (95, the edge probability i graph G (, K, P with P := P atifie q! ( K + 2 P. (35 From (28, we have K > K, which alog with (35 above yield The from (25 ad (36, it hold that q! ( K 2 P. (36 α = q l + (k l l! ( K 2 l + (k l l. (37 P A metioed, by lim α =, it hold that α 0 for all ufficietly large. The from (26, it follow that α = O(l l, (38 which alog with Lemma 7 ad 8, equatio (27 ad coditio P = Ω( iduce K = Ω ( (l 2. (39 Hece, we have lim K = ad it further hold for all ufficiet large that ( K 2 > ( K 2 3 ( K 2. (40 Applyig (40 to (37 ad the uig (27, Lemma 7 ad P = Ω(, it follow that α >! ( K ( K 2 l + (k l l P = α 3! Θ( P (l 2 = α O ( (l 2. (4 A oted at the begiig of Sectio 3, our proof i for 2 ice the cae of = already i proved by u 6. Uig 2 i (4, it hold that α > α + o(, which alog with (34 ad (38 yield (33. 29

30 6. Proof of Lemma 4 We firt how that the reult i true whe α i cofied a O(l l ad the demotrate that α ca be ideed cofied a O(l l. l +(k l l +α Firt, whe α i cofied a O(l l. From ρ = ad α = O(l l, l +(k l l +α±o( we ue Lemma 2 to obtai! t 2 P =. The we apply Theorem 2 ad the reult clearly follow i view of lim α ± o( = lim α. Secod, to how α ca be ideed cofied a O(l l, imilar to Lemma 6, we how the followig (a ad (b: (a For graph H (, t, P uder edge probability ρ = l + (k l l + β (42 with lim β =, there exit graph H (, t, P uder edge probability ρ = l + (k l l + β (43 with lim β = ad β = O(l l uch that there exit a graph couplig uder which H (, t, P i a paig ubgraph of H (, t, P. (b For graph H (, t, P uder edge probability ρ = l + (k l l + β (44 with lim β =, there exit graph H (, t, P uder edge probability ρ = l + (k l l + β (45 with lim β = ad β = O(l l uch that there exit a graph couplig uder which H (, t, P i a paig upergraph of H (, t, P. We firt prove property (a. We et ad P = P, (46 β = max{β, l l }. (47 Give (47 ad lim β =, we clearly obtai lim β = ad β = O(l l. It hold from (47 that β β, which alog with (42 (43 ad (46 yield t t. Uder t t ad P = P, by 26, Sectio 3, there exit a graph couplig uder which H (, t, P i a paig ubgraph of H (, t, P. We ow demotrate property (b. We et P = P, (48 30

31 ad β = mi{β, l l }. (49 Give (49 ad lim β =, we clearly obtai lim β = ad β = O(l l. It hold from (49 that β β, which alog with (44 (45 ad (48 yield t t. Uder t t ad P = P, by 26, Sectio 3, there exit a graph couplig uder which H (, t, P i a paig upergraph of H (, t, P. 3