Connected Balanced Subgraphs in Random Regular Multigraphs Under the Configuration Model

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1 Coected Balaced Subgraphs i Radom Regular Multigraphs Uder the Cofiguratio Model Liyua Lu Austi Mohr László Székely Departmet of Mathematics Uiversity of South Carolia 1523 Greee Street, Columbia, SC lu@mailbox.sc.edu, mohrat@ .sc.edu, laszlo@mailbox.sc.edu Abstract Our previous paper [9] applied a lopsided versio of the Lovász Local Lemma that allows egative depedecy graphs [5] to the space of radom matchigs i K 2, derivig ew proofs to a umber of results o the eumeratio of regular graphs with excluded cycles through the cofiguratio model [3]. Here we exted this from excluded cycles to some excluded balaced subgraphs, ad derive asymptotic results o the probability that a radom regular multigraph from the cofiguratio model cotais at least oe from a family of balaced subgraphs i questio. 1 The Tool I [9] we proved the followig theorem o extesios of partial matchigs that allows amog other thigs provig asymptotic eumeratio results about regular graphs through the cofiguratio model. Theorem 1 Let Ω be the uiform probability space of perfect matchigs i the complete graph K N N eve or the complete bipartite graph K N,N with N N. Let r rn be a positive iteger ad 1/16 > ɛ ɛn > 0 This research was supported i part by the NSF DMS cotract

2 as N approaches ifiity. Let M MN be a collectio of partial matchigs i K N or K N,N, respectively, such that oe of these matchigs is a subset of aother. For ay M M, let A M be the evet cosistig of perfect matchigs extedig M. Set µ µn M M PrA M. Suppose that M satisfies 1. M r, for each M M. 2. PrA M < ɛ for each M M. 3. M :A M A M PrA M < ɛ for each M M. 4. M:uv M M PrA M < ɛ for each sigle edge uv. 5. H M F Pr N 2r A H < ɛ for each F M. The, if rɛ o1, we have ad furthermore, if rɛµ o1, the Pr M M A M e µ+orɛµ, 1 Pr M M A M 1 + Orɛµ e µ. 2 I the theorem above PrA M deotes the probability accordig to the coutig measure, ad Pr N 2r A H idicates the probability of A H i a settig, whe 2r of the N vertices oe of them is a edpoit of a edge i the partial matchig H are elimiated, ad the probability is cosidered i this smaller istace of the problem. 2 The Cofiguratio Model ad the Eumeratio of d-regular Graphs For a give sequece of positive itegers with a eve sum, d 1, d 2,..., d d, the cofiguratio model of radom multigraphs with degree sequece d is defied as follows [3]. 1. Let us be give a set U that cotais N i1 d i distict miivertices. Let U be partitioed ito classes such that the ith class cosists of d i mii-vertices. This ith class will be associated with vertex v i after idetifyig its elemets through a projectio. 2. Choose a radom matchig M of the mii-vertices i U uiformly.

3 3. Defie a radom multigraph G associated with M as follows: For ay two ot ecessarily distict vertices v i ad v j, the umber of edges joiig v i ad v j i G is equal to the total umber of edges i M betwee mii-vertices associated with v i ad mii-vertices associated with v j. The cofiguratio model of radom d-regular multigraphs o vertices is the istace d 1 d 2 d, where d is eve. Beder ad Cafield [2], ad idepedetly Wormald, showed i 1978 that for ay fixed d, with d eve, the umber of d-regular graphs is 1 d 2 + o1 e 2 4 d d d e d d! Bollobás [3] itroduced probability to this eumeratio problem by defiig the cofiguratio model, ad brought the result 3 to the alterative form 1 + o1e 1 d2 4 d 1!! d!, 4 where the term 1 + o1e 1 d2 4 i 4 ca be explaied as the probability of obtaiig a simple graph after the projectio. The semifactorial d! d/2!2 d/2 d 1!! equals the umber of perfect matchigs o d elemets, ad d! is just the umber of ways matchigs ca yield the same simple graph after projectio. No-simple graphs, ulike simple graphs, ca arise with differet frequecies. Bollobás also exteded the rage of the asymptotic formula to d < 2 log, which was further exteded to d o 1/3 by McKay [10] i The strogest result is due to McKay ad Wormald [11] i 1991, who refied the probability of obtaiig a simple graph after the projectio to 1 + o1e 1 d2 4 d3 12 +O d2 ad exteded the rage of the asymptotic formula to d o 1/2. Wormald s Theorem 2.12 i [15] origially published i [14] asserts that for ay fixed umbers d 3 ad g 3, the umber of labelled d-regular graphs with girth at least g, is 1 + o1e P g 1 d 1 i d 1!! i1 2i d!. 5 [9] reproved the followig theorem of McKay, Wormald ad Wysocka [12] usig Theorem 1, uder a slightly stroger coditio tha d 1 2g 3 o i [12]: ote that a power of g i 6 oly restricts a secod term i the asymptotic series of the boud o g:

4 Theorem 2 I the cofiguratio model, assume d 3 ad g 6 d 1 2g 3 o. 6 The the probability that the radom d-regular multigraph has girth at least g 1 is 1 + o1 exp g 1 d 1 i i1 2i, ad hece the umber of d-regular graphs o vertices with girth at least g 3 is 1 + o1e P g 1 d 1 i d 1!! i1 2i d!. The case g 3 meas that the radom d-regular multigraph is actually a simple graph. Furthermore, the umber of d-regular graphs ot cotaiig cycles whose legth is i a set C {3, 4,..., g 1}, is 1 + o1e d 1 2 d 12 4 P d 1 i d 1!! i C 2i d!. This is a special case of a more geeral result. The followig defiitios are used i radom graph theory [1]. The excess of a graph G, deoted by κg, is EG V G. A graph G is balaced, if κh < κg for ay proper subgraph H with at least oe vertex. We first prove the followig Lemma. Lemma 3 Suppose that G is a coected balaced simple graph with κg 0. The the umber of subgraphs H with κh κg 1 is at most 2 V G 2. Proof: First we claim that G has o leaf vertex. Otherwise, if v is a leaf vertex, the κg v κg, a cotradictio. Let H be a subgraph of G with κh κg 1. If V H V G, the H is obtaied by deletig oe edge from G. The umber of such H s is EG. Now we assume V H V G. For ay vertex set S, let ΓS be the eighborhood of S i G. We defie a sequece of graphs H 0, H 1, H 2,... as follows. Let H 0 H. For i 1, if V H i 1 V G, we defie the graph H i as follows: V H i V H i 1 ΓV H i 1 ad EH i EH i 1 {uv : u V H i 1, v ΓV H i 1, ad uv EG}. Let H r be the last graph i the sequece. We have V H r V G. Observe κh κh 0 κh 1 κh 2 κh r κg. Sice κh κg 1, equalities hold for all but at most oe i the chai above. We have ΓV H i \ V H i 2 for all i r 1, as G has o leaf.

5 H H Figure 1: G H is either a ρ-shape or a path whe κh κg 1. It is easy to check that the differece of G ad H either forms a ρ-shape or is a path as show i Figure 1. A H with a ρ-shape may occur at most 2 EG times, a H with a path may occur at most most V G 2 times. Fially, EG + 2 EG + V G 2 2 V G 2. Let G be a family of coected balaced simple graphs with excess κ. We would like to estimate the probability that a radom d-regular multigraph cotais o graph i G. Give a simple graph G, let AutG be the umber of automorphisms of G. For ay k 2, let a k G be the umber of vertices with degree at least k. We defie a polyomial f G d k2 d k+1akg d 1 v V G d dv v 1 1! 1 d d v V V G G d dv v! d 1 2 EG V G. We have the followig theorem. Theorem 4 Let G be a family of coected balaced simple graphs with o-egative excess κ. Set r max EG. I the cofiguratio model, assume d 3 ad r 3 d 1 EG 1 o1 ad l r3 d κ+1 d 1 EG 1 2 o1. 7 The the probability that the radom d-regular multigraph arisig from the cofiguratio model cotais o subgraph i G is 1 + Ol exp f G d. AutG d κg Proof: Let ɛ Kr2 d 1 EG 1 with a large costat K. The first coditio makes sure rɛ o1, the secod coditio makes sure rɛµ Ol o1 i Theorem 1. For ay G G, let M G be the family of partial matchigs of U whose projectio is a copy of G. Suppose that G has s vertices v 1,..., v s ad t edges e 1,..., e t. For 1 i s, let C i be the class of d mii-vertices

6 associated to v i ad Q i be a ordered queue of d vi mii-vertices i C i. Let C be the parameter space of all possible C 1,..., C s, Q 1,..., Q s. We defie a mappig ψ : C M G as follows. For 1 j t, suppose that the edge e j has two ed-vertices v j1 ad v j2. We pop a mii-vertex x j from the queue Q j1, pop a mii-vertex y j from the queue Q j2, ad joi x j y j. Deote by M the collectio of edges {x j y j } 1jt. Clearly M forms a partial matchig whose projectio is a copy of G. We defie ψc 1,..., C s, Q 1,..., Q s M. Sice every partial matchig i M G ca be costructed i this way, ψ is surjective. For ay M M G ad ay C 1,..., C s, Q 1,..., Q s ψ 1 M, it uiquely determies a orderig of edges i M. The umber of such orderigs that give the same projectio G is exactly Aut G, the umber of edge automorphisms of G. By Whitey s Theorem [7], for a coected G, which is ot K 2 or K 1, Aut G AutG. We have ψ 1 M AutG. There are V G V G! ways to choose C1,..., C s. For 1 i s, there are d d dvi vi! ways to choose the queue Q i. We have Thus, C V G! V G v V G d d v d v!. M G C AutG 1 d V G! d v! AutG V G d v v V G f G d V G!d V G. 8 AutG V G For i 1, let G i be the set of graphs i G with exactly i edges. Let M i be the set of matchigs of U whose projectio gives a graph G G i ; there are exactly M G of them, ad they are couted i 8. The bad evets to be avoided are the projectio of some matchig from the uio M r i1 M i. For each M i M i i 1, 2,..., r, we have PrA Mi 1 d 1d 3 d 2i

7 We have µ M M r i1 i i1 i PrA M f G d V G V G!d AutG V G 1 d 1d 3 d 2i + 1 r f G d AutG d i V G 1 + O r 2 G 1 + O i 2 f G d AutG d κg. 10 d 1 EG +κ d κ d 1 EG κ. Observe from 10 that µ O O Now we verify the coditios of Theorem 1. Item 1 ad 2 are trivial by the defiitio of r ad ɛ. Item 3 ca be verified as follows. For two matchigs M ad M, A M A M if ay oly if M ad M coflict. For M M j, we have M :A M A M PrA M by symmetry argumet r i1 M M i:a M A M r i1 M M i 2r d r i1 2j d PrA M M M i PrA M Now we verify item 4. For ay uv M M, we have PrA M 2r d µ < ɛ. 11 M:uv M M PrA M < r i2 i r i2 i V G 2 V G 2!d 2 d v V G d dv v! d 1d 3 d 2i + 1 f G d d i V G O i 2 < ɛ. 12

8 1 We omitted a d 1 additive term from the estimate, which was there i [9], as it was there to hadle a loop. Fially, we have to verify item 5. For ay F M, we have to estimate M M F Pr N 2r A M. By the iequality below, this boils dow to estimatig M M F PrA M M M F Pr N A M, as with M i, Pr N 2r A M i d 2j 1 Pr N A M d r 2j 1 j1 i 2r 2r 1 + e 2 2r 2j 1 j1 d 4r 1. Assume that M M itersects F, M M \ F, ad the projectio of M is a graph G G. Let H be the projectio of F M. The graph H is a subgraph of G satisfyig 0 < EH < EG. Otherwise, G G cotradicts to the assumptio that G is balaced. We have PrA M M M F r i2 G G i H G EH r i2 G G i 1 + O V G i 2 r 2 G 1 + O V G V H dd 1 EG EH EG EH j1 d 2j + 1 H G EH G G H G EH d 1 EG EH κg κh d 1 EG EH. κg κh For the dd 1 base term i the secod lie, cosider that we ca build up G sequetially by always addig a edge icidet to a pre-existig compoet with at least oe edge, startig with the compoets of H with at least oe edge. Sice G is balaced, we have κg κh 1 for ay subgraph H with 0 < EH < EG. The last summatio ca be partitioed ito summatios over two classes. The first class C 1 cosists of H with κh κg 1. By Lemma 3, the umber of such H is at most V G 2. The secod class C 2 cosists of H with κh κg 2; there are most

9 2 EG of them. We boud d 1 EG EH by d 1 EG 1. We have PrA M 2 M M F < ɛ. G G H G EH 2 d 1 EG 1 G G Fially, the error i 10 does ot hurt, as e µ e 1+O e P e P ««r 2 G P f G d AutG d κg e O 1 Ole P d 1 EG EH κg κh «f G d AutG d κg «P r 2 G f G d AutG d κg 1 O f G d f G d AutG d κg, 2 V G 2 r 2 G f G d AutG d κg + 2 EG 2 f G d AutG d κg ad e P AutG d κg 1 + Ole µ. Corollary 4.1 We obtai Theorem 2 from Theorem 4, with the followig coditio, which is slightly weaker tha 6 i [9]: g 3 d 1 2g 3 o 13 Proof: Note that cycles are exactly the coected balaced graphs with κ 0. Let C 1 deote the graph of a oe-vertex loop ad C 2 the graph of a pair of multiedges betwee two vertices. These are balaced multigraphs with κ 0. Formally, we did ot allow i Theorem 4 balaced multigraphs, however, mior chages i the argumets will allow the iclusio of these two graphs see i [9] how to hadle loops ad parallel edges. The formulas exted for C 2 ad C 1, if we use as defiitio AutC 2 4 ad f C2 d d 1 2 ; AutC 1 2 ad f C1 d d 1. Applyig Theorem 4 to the family G C {C 1, C 2 }, where C {C 3,..., C g 1 } for g 3 oe obtais Theorem 2. Corollary 4.2 Uder the coditios of Theorem 4, the probability of obtaiig a balaced graph from G after projectio i the cofiguratio model,

10 i if ii if o1, ad the first part of 7 is stregth- d 1 EG 1 f o G d uiformly, is eed to r3 f G d AutG d κg 1 e P f G d AutG d κg is separated from zero, is f G d AutG d κg + Ole P f G d AutG d κg, d κ AutG f G d AutG d + O l + κg where l is little-oh of the mai term. f G d 2, AutG d κg Proof: i is straightforward. To obtai ii, use Ole x x + Ol + x 2 for l o1, x o1. The fact that l is little-oh of the mai term follows from the extra assumptio i ii. I the bipartite cofiguratio model we have two sets, U ad V, each cotaiig N mii-vertices, a fixed partitio of U ito d 1,..., d elemet classes, ad a fixed partitio of V ito δ 1,..., δ elemet classes. Ay perfect matchig betwee U ad V defies a bipartite multigraph with partite sets of size after a projectio cotracts every class to sigle vertex. I the regular case, d 1 d δ 1 δ d. We have the followig theorem Theorem 5 Let G be a family of coected balaced simple bipartite graphs with o-egative excess κ ad r max EG. I the regular case of the bipartite cofiguratio model, assume d 3 ad coditio 7. The the probability that the radom d-regular multigraph cotais o graph i G is 1 + o1e P 2f G d AutG d κg. Proof: We outlie the proof. For i 2,..., r, let M i be the set of matchigs of U ad V, whose projectio gives a graph G G with i edges. For a fixed G G, sice G is bipartite, let 1 G ad 2 G be the size of vertex partitio classes. The umber of matchigs, whose projectio is G, is exactly 2 AutG 1 G! 1 G 2 G 2 G! v V G d d v d v!. This formula is similar to Equatio 8. If G has o automorphism switchig its two colorclasses i particular whe 1 2, the we ca select

11 1 G classes from the classes of U ad select 2 G classes from the classes of V, or vice versa. This explais the costat factor 2. If G has a automorphism switchig its two colorclasses, the selectig 1 2 classes from U ad V, we obtai each copy of G Aut G /2 times from matchigs. The bad evets correspod to a matchig from the uio M r i1 M i. For each M i M i i 1, 2,..., r, we have We have M M r PrA M 2 i2 i 1 + O PrA Mi AutG 1G 2G r 2 G d 2i!. 14 d! v V G d d v d 2i! d v! d! 2f G d. 15 AutG d κg All the estimates go through as i the proof of Theorem 2. Applyig Theorem 5 to a family G {C 2 } C with C {C 4, C 6,..., C 2g 2 } a slight extesio to iclude C 2, like i [9], we get aother theorem of McKay, Wormald ad Wysocka [12], who actually had it without g 3 i 16. [9] reproved this theorem with g 6 i the coditio usig Theorem 1. Theorem 6 I the regular case of the bipartite cofiguratio model, assume that g is eve, d 3, ad g 3 d 1 2g 3 o. 16 The the probability that the radom bipartite d-regular multigraph does ot cotai a cycle of legth C {2, 4, 6,..., g 2}, is 1 + o1e P i C d 1 i i. Corollary 6.1 Corollary 4.2 applies to the bipartite regular cofiguratio model, chagig f G d to 2f G d. We are left with a ope problem of fidig asymptotics for the occurrece of a elemet of G ad obtaiig a simple multigraph simultaeously.

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