Between 2- and 3-colorability

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1 Betwee 2- ad 3-olorability Ala Frieze, Wesley Pegde Departmet of Mathematial Siees, Caregie Mello Uiversity, Pittsburgh PA 523. Jue 3, 204 Abstrat We osider the questio of the existee of homomorphisms betwee G,p ad odd yles whe p = /, < 4. We show that for ay positive iteger l, there exists ε = εl) suh that if = + ε the w.h.p. G,p has a homomorphism from G,p to C 2l+ so log as its odd-girth is at least 2l+. O the other had, we show that if = 4 the w.h.p. there is o homomorphism from G,p to C 5. Note that i our rage of iterest, χg,p ) = 3 w.h.p., implyig that there is a homomorphism from G,p to C 3. These results imply the existee of radom graphs with irular hromati umbers χ satisfyig 2 < χ G) < 2 + δ for arbitrarily small δ, ad also that 2.5 χ G, 4 ) < 3 w.h.p. Itrodutio The determiatio of the hromati umber of G,p, where p = for ostat, is a etral topi i the theory of radom graphs. For 0 < <, suh graphs otai, i expetatio, a bouded umber of yles, ad are almost-surely 3-olorable. The hromati umber of suh a graph may be 2 or 3 with positive probability, aordig as to whether or ot ay odd yles appear. For, we fid that the hromati umber χg, ) 3 with high probability, ad lettig k := sup χg, ) k, it is kow for all k ad k, k+ ) that χg, ) {k, k + }, see Luzak [7] ad Ahlioptas ad Naor [2]; for k > 2, the hromati umber may well be oetrated o the sigle value k, see Friedgut [5] ad Ahlioptas ad Friedgut []. Researh supported i part by NSF grat f030

2 I this paper, we osider fier otios of olorability for the graphs G, for, 3), by osiderig homomorphisms from G, to odd yles C 2l+. A homomorphism from a graph G to C 2l+ implies a homomorphism to C 2k+ for k < l. As the 3-olorability of a graph G orrespods to the existee of a homomorphism from G to K 3, the existee of a homomorphism to C 2l+ implies 3-olorability. Thus osiderig homomorphisms to odd yles C 2l+ gives a hierarhy of 3-olorable graphs ameable to ireasigly stroger ostrait satisfatio problems. Note that a fixed graph havig a homomorphism to all odd-yles is bipartite. Our mai result is the followig: Theorem. For ay l >, there is a ε > 0 suh that with high probability, G, +ε has odd-girth < 2l + or has a homomorphism to C 2l+. either Coversely, we expet the followig: Cojeture. For ay >, there is a l suh that with high probability, there is o homomorphism from G, to C 2l+ for l l. As 3 is kow to be at least 4.03, the followig ofirms Cojeture for a sigifiat portio of the iterval, 3 ). Theorem 2. For ay > 2.774, there is a l suh that with high probability, there is o homomorphism from G, to to C 2l+ for l l. We also have that l 4 = 2: Theorem 3. With high probability, G, 4 has o homomorphism to C 5. Note that as 3 > 4.03 > 4, we see that there are triagle-free 3-olorable radom graphs without homomorphisms to C 5. Our proof of Theorem 3 ivolves omputer assisted umerial omputatios. The same alulatios whih rigorously demostrate that l 4 = 2 suggest atually that l 3.75 = 2 as well. Our results a be reformulated i terms of the irular hromati umber of a radom graph. Reall that the irular hromati umber χ G) of G is the ifimum r of irumferees of irles C for whih there is a assigmet of ope uit itervals of C to the verties of G suh that adjaet verties are assiged disjoit itervals. Note that if irles C of irumferee r were replaed i this defiitio with lie segmets S of legth r, the this would give the ordiary hromati umber χg).) It is kow that χg) < χ G) χg), that χ G) is always ratioal, ad moreover, that χ G) p if ad oly if G has a homomorphism to q the irulat graph C p,q with vertex set {0,,..., q }, with v u wheever distv, u) := mi{ v u, v + q u, u + q v} q. See [9].) Sie C 2l+,l is the odd yle C 2l+ our results a be restated as follows: 2

3 Theorem 4. I the followig, iequalities for the irular hromati umber hold with high probability.. For ay δ > 0, there is a ε > 0 suh that, G = G, +ε odd girth 2. δ has χ G) 2 + δ uless it has 2. For ay > 2.774, there exists r > 2 suh that χ G, ) > r χ G, 4 ) < 3. Note that for ay ad l >, there is positive probability that G, has odd girth < 2l +, ad a positive probability that it does ot. I partiular, as the probability that G, has small odd-girth a be omputed preisely, Theorem gives a exat probability i 0, ) that G, +ε has a homomorphism to C 2l+. Ideed, Theorem implies that if = + ε ad ε is suffiietly small relative to l, the lim Prχ G, ) 2 + l+, 2 + l ]) = e φ l) e φ l+), ) where φ l ) = l i= 2i+ 22i + ). We lose with two more ojetures. The first oers a sort of pseudo-threshold for havig a homomorphism to C 2l+ : Cojeture 2. For ay l, there is a l > suh that G, has o homomorphism to C 2l+ for > l, ad has either odd-girth < 2l + or has a homomorphism to C 2l+ for < l. The seod asserts that the irular hromati umbers of radom graphs should be dese. Cojeture 3. There are o real umbers 2 a < b with the property that for ay value of, Prχ G, ) a, b)) 0. Note that our Theorem ofirms this ojeture for the ase a = 2. 2 Struture of the paper We prove Theorem i Setio 3. We first prove some strutural lemmas ad the we show, give the properties i these lemmas, that we a algorithmially fid a homomorphism. We prove Theorem 2 i Setio 4 by the use of a simple first momet argumet. We prove Theorem 3 i Setio 5. This is agai a first momet alulatio, but it has required umerial assistae i its proof. 3

4 3 Fidig homomorphisms Lemma. If α < /0 ad is a positive ostat where { } 6α < 0 = exp 3α the w.h.p. ay two yles of legth less tha α log i G,p, p =, are at distae more tha α log. Proof If there are two yles otraditig the above laim, the there exists a set S of size s 3α log that otais at least s + edges. The expeted umber of suh sets a be bouded as follows: 3α log ) s ) 2) ) 3α log s+ e ) s se ) s+ s+ s s + s 2 ) s=4 s=4 3α log 3α log s=4 < e2 ) 3α log log = o). e 2 2 ) s Our ext lemma is oered with yles i K 2 whih is the 2-ore of G,p. The 2-ore of a graph is the graph idued by the edges that are i at least oe yle. Whe >, the 2-ore osists of a liear size sub-graph together with a few vertex disjoit yles. By few we mea that i expetatio, there are O) verties o these yles. Let 0 < x < be suh that xe x = e. The w.h.p. K 2 has ν x) x ) verties ad µ x ) 2 2 edges. See for example Pittel [8]). If = + ε for ε small ad positive the x = η where η = ε + a ε 2, a 2 for ε < /0. The degree sequee of K 2 a be geerated as follows, see for example Aroso, Frieze ad Pittel [3]: Let λ be the solutio to λe λ ) e λ λ = 2µ ν x x = 2 + a ε + a ε. 4

5 We dedue from this that λ 4 a ε 8ε. We geerate the degrees d), d2),..., dν) as idepedet opies of the radom variable Z where for d 2, λ d PrZ = d) = d!e λ λ). We oditio that the sum D = d) + d2) + + d) = 2µ. We let θ k = Prdi) = d i, i =, 2,..., k D = 2µ) Prdi) = d i, i =, 2,..., k) = Prdk + ) + + d) = 2µ d + + d k ). Prd) + + d) = 2µ) It is show i [3] that if Z, Z 2,..., Z N are idepedet opies of Z the PrZ + + Z N = N EZ) t) = where σ 2 = Θ) is the variae of Z. σ 2πN + O )) t 2 + Nσ 2 2) We observe ext that the maximum degree i G,p ad hee i K 2 is q.s. at most log. It follows from this ad 2) that θ k = + o) for k log 2 ad θ k = O /2 ) i geeral. Lemma 2. For ay α, β, there exists 0 > suh that w.h.p. ay yle of legth greater tha α log i the 2-ore of G,p, p =, < < 0, has at most β log verties of degree 3. Proof Suppose that ) β 8εe e +8ε <. β We will show the that w.h.p. the K 2 does ot otai a yle C where i) C α log ad ii) C otais β C verties of degree greater tha two. We a boud the probability of the existee of a bad yle C as follows: I the followig display we hoose the verties of our yle i ν k) ways ad the arrage these verties i a yle C i k )!/2 ways. The we hoose βk verties to have degree at least three. We the sum over possible degree sequees for the verties i C. This explais the fator θ k λ d i k i= d i. We ow resort to usig the ofiguratio model of Bollobás [4]. This!e λ λ) would explai the produt k d i d i ) i=. We use the deomiator 2µ k to simplify the 2µ 2i+ alulatio. The ofiguratio model omputatio will iflate our estimate by a ostat A sequee of evets E is said to our quite surely q.s. if Pr E ) = O C ) for ay ostat C > 0. 5

6 fator that we hide with the otatio b. We write A b B for A = OB) whe OB) is ugly lookig. Pr C) b = ν k=α log ν k=α log ν k=α log ν k=α log ν k=α log ν k=α log = o). ν k 2k e k2 /µ 2k e k2 /µ 2k θ k 2k θ k 2k ) k )! 2 ) k θ k βk ν 2µ 2k)e λ λ) d,...,d βk 3 d βk+,...,d k 2 k λ d i d i!e λ λ) did ) i ) 2µ 2k i= ) k λ 2k k βk ) θ k d,...,d βk 3 d βk+,...,d k 2 ) k ) ν k λ 2k θ 2µe λ k e λ ) βk e β)kλ λ) βk ) k ) λ k θ e λ k e λ ) βk e β)kλ βk e k/µ e λ β λ e λ ) β ) ) β k e e λ β ) ) β k e e β)λ β k i= d i 2)! Lemma 3. For ay α ad ay k N, there exists ε 0 > 0 suh that w.h.p. we a deompose the edges of the G = G,p, p = +ε, 0 < ε < ε 0, as F M, where F is a forest, ad where the distae i F betwee ay two edges i M is at least k. Proof By hoosig β < i Lemma 2 we a fid, i every yle of legth > α log 2k of the 2-ore K 2 of G whih iludes all yles of G), a path of legth at least 2k + whose iterior verties are all of degree 2. We a thus hoose i eah yle of K 2 of legth > α log suh a path of maximum legth, ad let P deote the set of suh paths. Note that, i geeral, there will be fewer paths i P tha log yles i K 2 due to dupliates, but that the elemets of P are evertheless disjoit paths i K 2.) We ow hoose from eah path i P a edge from the eter of the path to give a set M. Note that the set of yles i G \ M is the same as the set of yles i G \ P P P. I partiular, the oly yles whih remai have legth α log ad are at distae k from M.) Thus, lettig M 2 osist of oe edge from eah yle of G \ M, Lemma implies that M = M M 2 is as desired. 6

7 Proof of Theorem. Our goal i this setio is to give a C 2l+ -olorig of G = G, +ε for ε > 0 suffiietly small. By this we will mea a assigmet : V G) {0,,..., 2l} suh that x y i G implies that x) y) as verties of C 2l+ ; that is, that x = y ± mod 2l + ). Cosider a deompositio of G as F M as give by Lemma 3, with k = 4l 2. We begi by 2-olorig F. Let F : V {0, } be suh a olorig. Our goal will be to modify this olorig to give a good C 2l+ olorig of S. Let B be the set of edges xy M for whih F x) = F y), ad let B be a set of distit represetatives for B, ad for i = 0,, let B i = {v B F v) = i}. We ow defie a ew C 2l+ olorig : V {0,,..., 2l}, by { F v) if dist F v, B) 2l v) = F x) ) j dist F x, v) + ) if x B j s.t. distx, v) F < 2l. 3) Color additio ad subtratio are omputed modulo 2l +.) Sie edges i M are separated by distaes 4l 2, this olorig is well-defied i.e., there is at most oe hoie for x). Moreover, is ertaily a good C 2l+ -olorig of F. Thus if is a ot a good C 2l+ -olorig of S, it is bad alog some edge xy M. But if suh a edge was already properly olored i the 2-olorig F, it is still properly olored by, sie it has distae 4l 2 2l from other edges i M. O the other had, if previously we had F x) = F y) = i, ad WLOG x B i, the the defiitio of v) gives that we ow have that x) {i, i + } modulo 2l ). Thus if is ot a good C 2l+ -olorig of S, the there is a edge xy M suh that x B i ad y s olor also hages i the olorig ; but by the distae betwee edges i M, this a oly happe if x ad y are at F -distae < 2l. Note also that F x) = F y) implies that dist F x, y) is eve. Thus i this ase, F {xy} otais a odd yle of legth 2l, ad so G has odd girth < 2l +, as desired. 4 Avoidig homomorphisms to log odd yles For large l, oe a prove the o-existee of homomorphisms to C 2l+ usig the followig simple observatio: Observatio 4. If G has a homomorphism to C 2l+, the G has a idued bipartite subgraph with at least V G) verties. 2l 2l+ Proof. Delete the smallest olor lass. 7

8 Proof of Theorem 2. The probability that G, verties is at most )2 β ) β 2 2 /4 < β has a idued bipartite subgraph o β ) 2 β e β2 /4 4) β β β) β The expressio iside the paretheses is uimodal i β for fixed, ad, for > 2.774, is less tha for β > I partiular, for > 2.774, G, has o homomorphism to C 2l+ for 2l +, 427, Avoidig homomorphisms to C 5 A homomorphism of G = G,p, p = ito C 5 idues a partitio of [] ito sets V i, i = 0,,..., 4. This partitio a be assumed to have the followig properties: P The sets V i, i = 0,,..., 4 are all idepedet sets. P2 There are o edges betwee V i ad V i+2 V i 2. Here additio ad subtratio i a idex are take to be modulo 5. P3 Every v V i, i =, 2, 3, 4 has a eighbor i V i. P4 Every v V 2 has a eighbor i V 3. Hatami [6], Lemma 2. shows that we a assume P,P2,P3. Give P,P2,P3, if v V 2 has o eighbors i V 3 the we a move v from from V 2 to V 0 ad still have a homomorphism. Furthermore, this move does ot upset P,P2,P3. We let V i = i for i = 0,,..., 4. For a fixed partitio we the have PrP P2) = p) S where S = PrP3 P P2) = PrP4 P P2 P3) ) 2 4 i i+. 5) i=0 4 p) i ) i. 6) i= 2 ) 3 p) 3) 2 7) Equatios 5) ad 6) are self evidet, but we eed to justify 7). Cosider the bipartite subgraph Γ of G,p idued by V 2 V 3. P3 tells us that eah v V 3 has a eighbor i V 2. Deote this evet by A. Suppose ow that we hoose a radom mappig φ from V 3 to V 2. We the reate a bipartite graph Γ with edge set E E 2. Here E = {xy : x V 3, y = φx)} ad E 2 is obtaied by idepedetly iludig eah of the 2 3 possible edges betwee V 2 ad V 3 with probability p. We ow laim that we a ouple Γ, Γ so that Γ Γ. 8

9 Evet A a be ostrued as follows: A vertex i v V 3 hooses B v eighbors i V 2 where B v is distributed as a biomial Bi 2, p), oditioed to be at least oe. The eighbors of v i V 2 will the be a radom B v subset of V 2. We oly have to prove the that if v hooses B v radom eighbors i Γ the B v stohastially domiates B v. But B v is oe plus Bi 2, p) ad domiatio is easy to ofirm. We have 2 istead of 2, sie we do ot wish to out the edge v to φv) twie. We ow write i = α i for i = 0,..., 4. We are partiularly iterested i the ase where = 4. Now 4) implies that G, 4 has o idued bipartite subgraph of size β for β > Thus we may assume that α i 0.06 for i = 0,..., 4. I whih ase we a write { ) } PrP P2 P3 P4) e o) 4 4 exp 2 α i α i+ e α i ) i) α The umber of hoies for V 0,..., V 4 with these sizes is ) ) = e o) 0,, 2, 3, i=0 αα i i Puttig α 4 = α 0 α α 2 α 3 ad b = b, α 0, α, α 2, α 3 ) = α 0 α 0α α α2 α 2α3 α 3α4 α 4 i=0 i= e α 3/α 2 e α 3 ) α 2. e α 0α 4 2 ) e α 0 ) α e α ) α 2 e α 2 ) α 3 e α 3 ) α 4 e α 3/α 2 e α 3 ) α 2, we see that sie there are O 4 ) hoies for 0,..., 4 we have Pr a homomorphism from G, 4 to C 5 ) e o) max α 0 + +α α 0,...,α b4, α 0, α, α 2, α 3 ). 8) I the ext setio, we desribe a umerial proedure for verifyig that the maximum i 8) is less tha. This will omplete the proof of Theorem 3. 6 Boudig the futio. Our aim ow is to boud the partial derivatives of b4.0, α 0, α, α 2, α 3 ), to traslate umerial omputatios of the futio o a grid to a rigorous upper boud. Before doig this we verify that w.h.p. G,p= 4 has o idepedet set S of size s = 3/5 or more. Ideed, Pr S) 2 p) 2) s 2 e 8/25 e 2/5 = o). 9

10 I the alulatios below we will make use of the followig bouds: They assume that 0.06 α i 0.6 for i 0. logα i ) > 2.82;.3 < loge 4α i ) < 2.3; e 4α i < 3.69; loge α 3/α 2 +4α 3 ) > 0.9; e 4α i e 4α i < α 2 e α 3/α 2e 4α 3 < We ow use these estimates to boud the absolute values of the b b α i. Our target value for these is 30. We will be well withi these bouds exept for i = 2 Takig logarithms to differetiate with respet to α 0, we fid b = b, α 0, α, α 2, α 3 ) α 0 α 0 + α + α ) e α 0 + α 4 I partiular, for = 4, ) logα 0 ) + logα 4 ) loge α3 ). 9) b b 4α 0 + logα 4 ) loge 4α 3 ) > , α 0 b b 4 α + α ) α 0 e α 0 + α 4 logα 0 ) loge 4α 3 ) < Similarly, we fid b = b, α 0, α, α 2, α 3 ) α α 0 + α 2 + α ) 2 e α ad so for = 4, logα ) + logα 4 ) + log e α 0 )), 0) e α 3 b b 4α 0 + logα 4 ) + loge 4α 0 ) loge 4α 3 ) > , α b b 4 α 2 + α ) 2 logα α e 4α ) loge 4α 3 ) < We ext fid that b = b, α 0, α, α 2, α 3 ) α 2 α 0 + α 3 + α ) 3 α 3 /α 2 e α 2 e α 3/α 2 +α 3 + log α 4 log α 2 + loge α ) loge α 3 ) α 3 α 2 α 3 loge α 3/α 2 +α 3 ); ) 0

11 ad so for = 4, b b α 2 4α 0 α 3 α 2 e α3/α2+α3 e α 3/α 2 +α 3 loge α 3/α 2 +α 3 ) + logα 4 ) + log e 4α ) e 4α 3 We eed to be a little areful here. Now α 3 /α 2 0 ad if α 3 /α 2 9 the α ad the α i = 0.28 for i 3. We boud b b α i Cotiuig we get α 3 9 : α 2 b b > = 29.97, α 2 b α 3 for both possibilities. 9 : α 2 b > , α 2 b 4 α 3 + α ) 3 e 4α ) logα α 2 e 4α 2 ) + log loge α 3/α 2 +α 3 ) 2 e 4α 3 b Fially, we fid that < b = b, α 0, α, α 2, α 3 ) α 3 ) e α 3 α 0 + α 4 + e α 3 + α 2 e α 3/α 2e α 3 + logα 4 ) logα 3 ) + log e α 2 ) e α 3 ad so for = 4 b b 4α 0 + logα 4 ) + loge 4α 2 ) loge 4α 3 ) > , α 3 b b e 4α3 + 4α 2 e 4α 2 ) 4α 4 + α 3 e 4α 3 e α 3/α 2e 4α 3 logα 3 ) + log e 4α 3 < We see that b b α i < 30 for all 0 i 3. Thus, if we kow that b, α 0, α, α 2, α 3 ) B for some B, this meas that we a boud b4, α 0, α, α 2, α 3 ) < ρ by hekig that b4, α 0, α, α 2, α 3 ) < ρ ε o a grid with step-size δ ε/2 B 30). The C++ program i Appedix A heks that b4, α 0, α, α 2, α 3 ) <.949 o a grid with step-size δ =.0008 it ompletes i aroud a hour or less o a stadard desktop omputer, ad is available for dowload from the authors websites). Suppose ow that B is the supremum of b4, α 0, α, α 2, α 3 ) i the regio of iterest. For ε = 60δB = 0.048B, we must have at some δ-grid poit that b4, α 0, α, α 2, α 3 ) B ε =.962B.962. This otradits the omputer-assisted boud of <.949 o the grid, ompletig the proof of Theorem 3. 2) Referees [] D. Ahlioptas ad E. Friedgut, A sharp threshold for k-olorability, Radom Strutures ad Algorithms 4 999)

12 [2] D. Ahlioptas ad A. Naor, The two possible values of the hromati umber of a radom graph, Aals of Mathematis ), [3] J. Aroso, A. Frieze ad B. Pittel, Maximum mathigs i sparse radom graphs: Karp-Sipser revisited, Radom Strutures ad Algorithms 2 998), -78. [4] B. Bollobás, A probabilisti proof of a asymptoti formula for the umber of labeled graphs, Europea Joural o Combiatoris 980) [5] E. Friedgut, Sharp Thresholds of Graph Properties, ad the k-sat Problem, Joural of the Ameria Mathematial Soiety 2 999) [6] H. Hatami, Radom ubi graphs are ot homomorphi to the yle of size 7, Joural of Combiatorial Theory B ) [7] T. Luzak, A ote o the sharp oetratio of the hromati umber of radom graphs, Combiatoria 99) [8] B. Pittel, O Tree Cesus ad the Giat Compoet i Sparse Radom Graphs, Radom Strutures ad Algorithms 990) [9] X. Zhu, Cirular hromati umber: a survey, Disrete Mathematis ) Xudig Zhu 2

13 A C++ ode to hek futio boud #ilude <iostream> #ilude <math.h> #ilude <stdlib.h> usig amespae std; it maiit arg, har* argv[]){ double delta=.0008; //step size double maxidset=.6; //o idepedet sets larger tha this fratio double miclass=.06; //all olor lasses larger tha this fratio double val=0; double maxval=0; double maxa0,maxa,maxa2,maxa3; //to reord the oordiates of max value maxa0=maxa=maxa2=maxa3=0; double A23,A,B,C; //For preomputig parts of the futio double =4; for double a3=miclass; a3 + 4*miClass<; a3+=delta){ B=exp*a3)-; for double a2=miclass; a3 + a2 + 3*miClass<; a2+=delta){ A23=/powa2,a2)*powa3,a3)) * exp-/2) * powexp*a2)-,a3) * pow-exp-a3/a2)*exp-*a3),a2); for double a=miclass; a3+a<maxidset && a3 + a2 + a + 2*miClass<; a+=delta){ A=A23/powa,a)* powexp*a)-,a2); for double a0=maxmaxmiclass,.4-a2-a3),.4-a-a3); a2+a0<maxidset && a3+a0<maxidset && a3 + a2 + a + a0 + miclass<; a0+=delta){ double a4=-a0-a-a2-a3; C=exp*a0); val=/powa0,a0) * A * powb*c/a4,a4)* powc-,a); if val>maxval){ maxval=val; maxa0=a0; maxa=a; maxa2=a2; maxa3=a3; } } } } } out << "Max is "<<maxval<<", obtaied at " <<maxa0<<","<<maxa<<","<<maxa2<<","<<maxa3<<"," <<-maxa0-maxa-maxa2-maxa3<<")"<<edl; } 3

14 program output: $./boud Max is , obtaied at ,0.2568,0.704,0.632,0.92) 4

Between 2- and 3-colorability

Between 2- and 3-colorability Betwee 2- ad 3-colorability Ala Frieze ad Wesley Pegde Departmet of Mathematical Scieces Caregie Mello Uiversity Pittsburgh PA 523, U.S.A ala@radom.math.cmu.edu, wes@math.cmu.edu Submitted: Sep 0, 204;