Dstrbuton of subgraphs of random regular graphs Zhcheng Gao Faculty of Busness Admnstraton Unversty of Macau Macau Chna zcgao@umac.mo N. C. Wormald Department of Combnatorcs and Optmzaton Unversty of Waterloo Waterloo ON Canada N2L 3G1 nwormald@uwaterloo.ca 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
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
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
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
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
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
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, 2000. [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) 213 223. [5] A. Rucńsk, When are small subgraphs of a random graph normally dstrbuted? Probablty Theory and Related Felds 78 (1988), 1 10. 7