Distribution of subgraphs of random regular graphs

Transcription

1 Dstrbuton of subgraphs of random regular graphs Zhcheng Gao Faculty of Busness Admnstraton Unversty of Macau Macau Chna N. C. Wormald Department of Combnatorcs and Optmzaton Unversty of Waterloo Waterloo ON Canada N2L 3G1 1 Introducton The asymptotc dstrbuton of small subgraphs of a random graph has been bascally worked out (see Rucńsk [5] for example). But for random regular graphs, the man technques for provng, for nstance, asymptotc normalty, do not seem to be usable. One very recent result n ths drecton s to be found n [3], where swtchngs were appled to cycle counts. The am of the present note s to show that another very recent method of provng asymptotc normalty, gven by the authors n [1], can easly be appled to ths problem. In partcular, t requres consderably less work than usng swtchngs. The applcaton s, however, not drect, n the sense that the result obtaned s very weak f the random varable countng copes of a subgraph s examned drectly. We obtan a much stronger result by consderng solated copes of a subgraph. To be specfc, we nvestgate the probablty space G n,d of unformly dstrbuted random d-regular graphs on n vertces (whch we assume to be {1, 2,..., n}). Asymptotcs are for n, and here d s not fxed but may vary wth n (though for all our results there s an upper bound on the growth of d, at least mplctly). As usual, we mpose the restrcton that for the asymptotcs, the odd values of n are omtted n the case of odd d. Research supported by NSERC and Unversty of Macau Research supported by the Canada Research Chars program and NSERC. 1

2 We use µ(g) and ν(g) for the numbers of edges and vertces of a graph G respectvely. A graph G s strctly balanced f µ(g) ν(g) > µ(g 1) ν(g 1 ) (1.1) for all nontrval proper subgraphs G 1 of G. A standard example: every connected regular graph s strctly balanced. Throughout ths paper, [x] m denotng the fallng factoral: x(x 1) (x m + 1). We wll use the followng from [1] to deduce asymptotc normalty. Theorem 1 Let s n > µ 1 n and σ n = µ n + µ 2 ns n, (1.2) where 0 < µ n. Suppose that µ n = o(σ 3 n), (1.3) and a sequence {X n } of nonnegatve random varables satsfes ( k E[X n ] k µ k 2 ) s n n exp 2 (1.4) unformly for all ntegers k n the range cµ n /σ n k c µ n /σ n for some constants c > c > 0. Then (X n µ n )/σ n tends n dstrbuton to the standard normal as n. We also use McKay [4, Theorem 2.10], n the form of the followng smpler specal case stated n [3]. Here, G denotes a random element of G n,d, E denotes the edge set, and K n s the complete graph on n vertces (the same vertex set as G). Theorem 2 For any d and n such that G n,d 0, let J E(K n ). Then, wth j the number of edges n J ncdent wth vertex, (a) f J + 2d 2 nd/2 then P(J E(G)) (b) f 2 J + 4d(d + 1) nd/2 then nk=1 [d] jk 2 J [nd/2 2d 2 ] J ; P(J E(G)) nk=1 [d] jk 2 J [nd/2 1] J ( ) J n 2d 2. n + 2d Corollary 1 Provded d J = o(n), the hypotheses of Theorem 2 mply that P(J E(G)) = nk=1 [d] jk (nd) J ( 1 + O ( (d J /n) 2 )). 2

3 2 Dstrbuton of number of copes of a graph Throughout ths secton, G denotes a random graph n G n,d. Let H be a fxed strctly balanced graph wth maxmum vertex degree d. Let p and q be the number of vertces and edges of H. A copy of H n G s a subgraph of G whch s somorphc to H. The use of Theorem 1 calls for computng hgh moments of a random varable. It turns out that the random varable countng copes of H has badly behaved moments and consequently does not produce a very useful result. Instead we consder a related random varable whose behavour s more easly analysed. We say that a subgraph of a graph G somorphc to H s an solated copy of H f t shares no edges wth any other subgraph of G somorphc to H. Let X H be the random varable whch s the number of solated copes of H n a random d-regular graph. Let a denote the order of the automorphsm group of H. Set µ = P(H 1 G)[n] p /a (2.1) where H 1 s a fxed copy of H on the vertex set {1, 2,..., V (H)}. The probablty that any gven copy of H n K n occurs n G s equal to P(H 1 G), and there are [n] p /a such copes. Hence, µ s the expected number of copes of H n G, solated or not. By Corollary 1, for d = o(n) (notng that q s fxed), where f = Θ(g) f f = O(g) and g = O(f). Also, let µ = Θ(n p q d q ) (2.2) r = r(n, d, H) = P(H 1 s not solated H 1 G). (2.3) Fx a proper subgraph F of H 1 contanng at least one edge, and consder the probablty that G contans not only H 1 but also the edges of a subgraph H 2 = H wth H1 H 2 = F. Agan usng Corollary 1, ths (uncondtonal) probablty s Θ(n p p(f ) (d/n) 2q q(f ) ) for d = o(n). Snce there s a bounded number of such subgraphs F, we have for d = o(n), r = Θ(r H (n, d)) where r H (n, d) = n p p(f ) ( d n) q q(f ), (2.4) p(f ) = V (F ), q(f ) = E(F ), and F s a subgraph of H whch maxmses n q(f ) p(f ) d q(f ) subject to 1 q(f ) < q. (See [2, Secton 3.2] for a related dscusson n the settng of random graphs wthout the regularty condton.) Theorem 3 Defne µ and r as n (2.1) and (2.3). Suppose that µ, µ = o(n), µ = o(n 2 /d 2 ) and r = O(1/ µ). Then (X H µe r )/σ tends n dstrbuton to the standard normal as n, where σ 2 = µe r. Note 1 If r = o(1/ µ) then the mean and varance of the asymptotc dstrbuton can both be taken as µ. Moreover, the proof of the theorem then smplfes consderably. However, by ncludng the case r 1/ µ we hghlght why the method does not easly extend. 3

4 Note 2 The dstrbuton result n [3], whch s only for cycles, does not extend to the full range of d covered by Theorem 3. (It does however apply to non-fxed subgraphs, a modfcaton whch could also be done easly usng the technques of the present paper.) One could presumably extend the methods used n [3] to obtan dstrbuton results for all strctly balanced subgraphs, but ths s not as economcal as the method n the present paper, and we beleve that the range of d obtaned would not be any greater than that n Theorem 3. Note 3 Dstrbuton results for subgraph counts n the other common models of random graphs apply for wder ranges of densty of the parent graph than expressed n Theorem 3. Gven the much greater accessablty of those models due to edge ndependence, ths s not very surprsng. Proof of Theorem 3 We compute the k th factoral moment E[X H ] k, for k = O( µ), k. Note that ( ) E[X H ] k = P A J (2.5) J 1,...,J k E(K n) where A J denotes the event that J E(G) and forms an solated copy of H. (For k = 1, ths dffers from (2.1) because the copes here are solated.) To fnd the number of nonzero summands contrbutng n (2.5), for whch a prerequste s that the J are parwse dsjont, consder placng k ordered copes of H on the vertces of the complete graph. Snce k = O( µ) = o( n), the number of ways of dong ths, where each copy s placed ndependently (gnorng possble overlaps) s asymptotc to n kp a k, (2.6) and we get the same expresson f we nsst that the copes have dsjont vertex sets (so after j copes have been placed there are n pj vertces to choose from). Thus, by sandwchng, ths s also asymptotcally the number of ways of choosng edge-dsjont copes, as requred for solated copes, and almost all these placements are parwse vertex-dsjont. Clearly ( ) P A J P(J 1 J k E(G)) (P(J 1 G)) k (1 + o(1)) usng Corollary 1 and notng that the assumpton µ = o(n 2 /d 2 )) mples the requred bound on d J. So we have from (2.5) that ( ) n kp E[X H ] k = o (P(J a k 1 G)) k + = o ( ) n kp a k ( ) P A J ( ) A J J 1,...,J k E(Kn) J vertex-dsjont (P(J 1 G)) k + nkp a k P (2.7) for any partcular choce of J 1,..., J k E(K n ) whch nduce vertex-dsjont copes of H n K n. Lettng B denote the condtonal probablty that these sets nduce solated 4

5 copes n G, gven that they are subsets of the edge set of G, we have ( ) P A J = B P(J 1,..., J k E(G)) B(P(J 1 G)) k (2.8) usng Corollary 1 agan. For one of the copes to be nonsolated, t must share an edge wth some other copy of H, and we may use the same machnery to compute the factoral moments of the number Y of for whch J nduces a nonsolated copy n G, condtonal upon J 1,..., J k E(G). We have kr = O(1) (2.9) snce r µ = O(1), and hence also, snce µ, r = o(1). (2.10) Thus r = O(r 2 ) + H 2 P(H 1 H 2 G) P(H 1 G) where the sum s over copes H 2 of H n K n whch share at least one edge wth H 1, and the r 2 term bounds the overcountng n ncluson-excluson. Wth ths observaton t s easy to compute the jth factoral moment E[Y ] j of Y. There are [k] j ways to choose a j-subset of J 1,..., J k each of whch nduce a copy of H to be nonsolated, and the probablty that all the requred edges are present n G s, agan usng Corollary 1, asymptotc to r j P(J 1,..., J k E(G)) r j (P(J 1 G)) k. The probablty that all these edges are n G, condtonal upon the event that J 1,..., J k E(G), s thus r j, and so E[Y ] j (kr) j. Thus by (2.9) and the method of moments (the usual one, that s), P(Y = 0) e kr. Thus B e kr, and we have from (2.7) and (2.8) that E[X H ] k nkp a k e kr (P(J 1 G)) k. By (2.1) and the fact that k = o( n), ths mples E[X H ] k (µe r ) k and thus by Theorem 1 wth s n = 0 the dstrbuton of X H s asymptotcally normal wth mean and varance µe r by (2.10). Example: Cycles Consder the graph H = C t, the cycle of length t 3. Here by (2.2) µ = Θ(d t ), so we requre d. Also t s easy to check that r H (n, d) = Θ(µ/(nd)), the maxmum n (2.4) occurrng for F = K 2. Thus the range of µ s bounded at the maxmum end by µ 3/2 = O(nd), µ = o(n) and µ = o(n 2 /d 2 ). By consderng the mpled upper bounds on d, we see that the frst s strctest, and thus the number of solated copes of H s 5

6 asymptotcally normally dstrbuted provded d and d = O(n 2/(3t 2) ). Ths can be compared wth the result n [3], for whch the bound s d = o(n 1/(2t 1) ). Example: Complete graphs Consder the graph H = K t, where t 3. Here by (2.2) µ = Θ(d t(t 1)/2 n t(t 3)/2 ). (2.11) Also r H (n, d) = max 2 s<t d t(t 1)/2 s(s 1)/2 n t(t 3)/2+s(s 3)/2, and consderng d = n α, the maxmum occurs (as wth cycles) at s = 2, and so r = Θ(µ/(nd)). Thus (as wth cycles), the range of µ s bounded at the maxmum end by µ 3/2 = O(nd), µ = o(n) and µ = o(n 2 /d 2 ). By consderng the mpled upper bounds on d, t s straghtforward to verfy that the frst gves the strctest bound for t = 3, and the last does for t 4. These mply the upper bounds d = O(n 2/7 ) n the case t = 3, and d = o(n (t(t 3)/2+2)/(t(t 1)/2+2) ),.e. d = o(n 1 2t/(t2 t+4) ), for t 4. Therefore, the number of solated copes of H s asymptotcally normally dstrbuted provded ths upper bound on d holds and the expresson n (2.11) tends to. Fnally, we may conclude somethng about the dstrbuton of the total number of copes of H, solated or not. Denote ths number by ˆX H. Corollary 2 Suppose that µ, µ = o(n), µ = o(n 2 /d 2 ) and r = o(1/ µ). Then ( ˆX H µ)/ µ tends n dstrbuton to the standard normal as n. Proof: The expected number of nonsolated copes s O(µr). So we may conclude that the total number of copes of H s asymptotcally normal provded µr = o( µ),.e. r µ = o(1). Ths s an assumpton of the corollary whch s stronger than the correspondng one n the theorem. 3 Concludng remarks For the dstrbuton results obtaned n Theorem 3, the mean and varance are asymptotcally equal. Ths means that t could equvalently be stated as gvng asymptotcally Posson dstrbuton. It would be nterestng to know the range of the degree d for whch the subgraph count remans asymptotcally Posson. Theorem 1 can potentally be used to deduce asymptotc normalty outsde the Posson range (as for nstance the prevous applcatons n [1]); one challenge s to fnd a way to apply t for such d n the present context. Another challenge s to fnd a way to apply any of the other methods of deducng asymptotc normalty to sgnfcantly hgher values of d than we do here. One possblty s to use swtchngs rather than standard ncluson-excluson to extend the range of d for whch the nonsolated copes may be treated n the proof of Theorem 3. However, the extra effort may not pay very bg dvdends. 6

7 References [1] Z. Gao and N.C. Wormald, Asymptotc normalty determned by hgh moments, and submap counts of random maps, Probablty Theory and Related Felds (to appear). [2] S. Janson, T. Luczak and A. Rucńsk, Random graphs, Wley, New York, [3] B.D. McKay, N.C. Wormald and B. Wysocka, Short cycles n random regular graphs, Electronc Journal of Combnatorcs (submtted). [4] B. D. McKay, Subgraphs of random graphs wth specfed degrees, Congressus Numerantum 33 (1981) [5] A. Rucńsk, When are small subgraphs of a random graph normally dstrbuted? Probablty Theory and Related Felds 78 (1988),