Topics in Graph Automorphisms

Transcription

1 Topcs n Graph Automorphsms Derrck Stolee Unversty of Nebraska-Lncoln s-dstolee1@math.unl.edu February 15, 2010 Abstract The symmetry of a graph s measured by ts automorphsm group: the set of permutatons of the vertces so that all edges and non-edges are preserved. There are natural questons whch arse when consderng the automorphsm group and there are several nterestng results n ths area that are not well-known. Ths talk presents some of these results and maybe even proves one or two. Frst, we wll prove that almost all graphs have trval automorphsm group. Second, we wll brefly dscuss the relaton of the graph automorphsm and group ntersecton problems. Then, we wll dscuss Baba s constrcuton that any fnte group wth n elements can be represented by a graph on 2n vertces (other than three exceptons). Fnally, we wll menton there exsts a subgroup of S n that s the automorphsm group of no graph of sze less than 1 2 ( n 1 2 n). 1 Almost all graphs are rgd Before we can begn the proof of ths fact, recall the Chernoff-Hoeffdng bounds. Theorem 1.1 ([DP09]). Let X = n =1 X be a sum of dentcally dstrbuted ndependent random varables X where Pr(X = 1) = p, Pr(X = 0) = q = 1 p. Then, we have the followng relatve Chernoff-Hoeffdng bound for all ε > 0: Pr[X < (1 ε)np] e npε2 /2, Pr[X > (1 + ε)np] e npε2 /2 1.1 Propertes of G(n, p) Lemma 1.2. Let ε be a functon on n wth ε(n) > 0. Then, the probablty that G, dstrbuted as G(n + 1, p), has all vertces of degree deg v ((1 ε)np, (1 + ε)np) s at least 1 2(n + 1)e npε2 2. Proof. Let X,j be the ndcator varable for the edge {, j} appearng n G(n + 1, p) (1 < j n + 1). The expected value s p. By lnearty of expectaton, E[deg ] = E[X,j ] = np. j By the Chernoff bound, And smlarly, Pr[deg < (1 ε)np] e npε2 /2. Pr[deg > (1 + ε)np] e npε2 /2. 1

2 Hence, Pr[deg ((1 ε)np, (1 + ε)np)] 2e npε2 /2. Thus, the probablty that all vertces have degree wthn the requested bounds s at least 1.2 Rgdty of G(n, p) 1 2e npε2 /2 = 1 2(n + 1)e npε2 2. Theorem 1.3 ([Bol01]). Let G G n,p for constant p and let ε > 0. The graph G s rgd wth probablty whch tends to 1 as n tends to nfnty. Pr[Aut(G) I] 1 2ne (n 1)pε2 /2 n (n 1)p(1 ε), Proof. Consder G havng the property that all vertces v V(G) have degree bounds (1 ε)np deg v (1 + ε)np. If G has automorphsm group Γ S n, then Γ acts on the pars of vertces. Ths parttons them n to par orbts that ether consst entrely of edges or have no edges at all. If ths partton has k parts, then at most 2 k graphs have ths partton and as such has automorphsm group Γ. We wll argue that f Γ s non-trval, then k s at most ( n 2 ) (n 1)p(1 ε) + 1. Most clearly, a non-trval group has a partton σ that moves some vertex x to a dfferent vertex y. By ths acton, the edges of x are mapped to edges of y. These edges are therefore not n sngleton parttons (except possbly for the edge xy f t exsts). But snce there are at least (n 1)p(1 ε) 1 edges ncdent to y that do not form ther own orbts, the number of orbts s at most ( n 2 ) (n 1)p(1 ε) + 1. So, the portons of graphs wth an automorphsm that takes x to y s at most 2 (n 2 ) (n 1)p(1 ε)+1 2 (n 2 ) = 2 1 (n 1)p(1 ε). The sum over all pars x, y gves the porton wth non-trval automorphsms s at most n (n 1)p(1 ε) (usng ( n 2 ) n2 ). Ths probablty s condtoned on the bounded degree. Removng the condton gves a probablty of rgdty beng at least [ 1 2ne (n 1)pε 2 /2 ] [1 n (n 1)p(1 ε)] = 1 2ne (n 1)pε2 /2 n (n 1)p(1 ε) + 2n 2 e ln 2 (n 1)p[ε2 /2+ln 2(1 ε)] 1 2ne (n 1)pε2 /2 n (n 1)p(1 ε). 2 Group Intersecton and Graph Automorphsm The problem of fndng the ntersecton of two groups s at least as hard as fndng the automorphsm group of a graph. Showng the reverse hardness s very dffcult, f possble at all. Ths secton defnes these problems and dscuss how hardness may be shown. 2.1 Group Intersecton Let X be a set of elements n S m, the permutaton group on ms elements. Ths set X generates a group X as the set of all fnte products of elements n X and ther nverses. Note that ths ncludes the empty product whch s taken to be the dentty element. Here, X s a generatng set and the group X s the group generated by X. Gven two generatng sets X 1, X 2, the groups X 1 and X 2 ntersect at least at the dentty element. Checkng f the ntersecton s trval defnes the followng decson problem. 2

3 Defnton 1 ([Hof82]). The decson problem of Group Intersecton s the language GrpInt D defned as GrpInt D = {[0 m, X 1, X 2 ] : X 1 X 2 {ε}}. Here, the nput s nterpreted as generatng sets X 1, X 2 S m and ts sze s n = m + ( X 1 + X 2 ) log m. A polynomal on the nput of GrpInt D s taken as a functon n poly(m, X 1, X 2 ). Ths decson problem has a smple NPalgorthm: non-determnstcally choose a non-trval element x n S m and use the polynomal-tme algorthm for group ownershp to check that x X 1 and x X 2. If x n both, accept. Whle the decson problem has a clear NPalgorthm, the problem of producng explct generators X 3 that generate X 1 X 2 seems much harder. Defnton 2. Producng generators for the ntersecton of a group s the problem GrpInt P : wth the condton that X 3 = X 1 X 2. [0 m, X 1, X 2 ] GrpInt P [X 3 ], The smplest non-determnstc algorthm for ths problem requres alternaton. Frst, non-determnstcally choose X 3 and test that X 3 X 1 X 2. Ths can be done usng an NP process. However, to show that X 3 X 1 X 2, an element x of S m must be chosen non-determnstcally, verfed to be n X 1 X 2, and then verfed to be n X 3. If the frst verfcaton fals, return falure (not reject or accept). If the frst verfcaton succeeds, reject f the second verfcaton fals. Ths s a conp process as all choces of x wll return falure or accept f X 3 generates the ntersecton, but at least one path wll reject f t does not. Hence, GrpInt P NP conp. 2.2 Graph Automorphsm Gven a graph G, the automorphsm group Aut(G) s the set of permutatons on the vertex set V(G) that preserve adjacences and non-adjacences. Explctly, Aut(G) = {σ Sym(V(G)) : v, u V(G), {σ(v), σ(u)} E(G) {v, u} E(G)}. Ths gves an mmedate decson problem on all graphs. Defnton 3 ([Hof82]). The decson problem of Graph Automorphsm s GA D defned as GA D = {[G] : Aut(G) {ε}}. Here, the nput s a graph G gven as an adjacency matrx. The nput sze s the number of vertces n = V(G), even though the encodng s sze n 2. Agan, ths problem s clearly n NP. Non-determnstcally choose a non-trval element σ Sym(V(G)) and check f t preserves adjacences and non-adjacences n G. If so, σ Aut(G) {ε}. Ths leads drectly to the producton verson of the problem. Defnton 4. Producng generators for the automorphsm of a graph s the problem GA P : where X = Aut(G) Sym(V(G)). [G] GA P [X], Note that these defntons are very smlar to those of GrpInt D and GrpInt P. Also, t seems GrpInt D has the same upper bound on ts complexty. However, GA P has a dfferent complexty. In fact, GA P s polynomal-tme Turng reducble to the decson problem of graph somorphsm, GI D. 3

4 2.3 Reductons A many-one reducton from a decson problem A to a decson problem B s a functon f : Σ Σ so that f (x) B f and only f x A. Ths can be nterpreted as an nput manpulaton functon. Propose that a solver for B s gven. The functon f can be used to convert the nput x for problem A nto nput f (x) sutable for the B-solver. Ths solver then computes an answer for f (x) on B whch s the correct answer for x on A. Ths proves that the complexty of A s at most the complexty of f plus the complexty of B. If f s computed n log-space, then the orderng of ther complexty s denoted A L m B. If f s computed n polynomal tme, then the orderng s denoted A P m B. A many-one reducton from a producton problem A to a producton problem B s a par of functons f, g : Σ B Σ so that f (x) y mples x A g(y). The functon f acts as an nput converter, smlar to the decson reducton. However, the outputs of a B-solver may not ft exactly the format of an A-solver, so the functon g converts the B output nto format ft for A. Hence, f a B-solver exsts, the followng chan wll solve A: B x f (x) y g(y) A The complexty of the functon f wll gve the same orderngs L m and P m as before. The use of the functon g could be a method for cheatng. Instead of allowng computaton to be done n g, t wll need to be consdered a well-defned njecton (wth respect to equvalence classes on both sdes of the functon). Ths enforces that the reducton needs to use the problem B to produce an answer that almost mmedately gves an answer to A. Consder the followng theorem as a common use of these reductons. Theorem 2.1 ([Hof82]). GA P L m GrpInt P. Rdculously poor sketch of proof. f ([G]) = [0 (n 2 ), Sym(V(G)), Sym(E(G)) Sym(E(G))]. g : Sym(V(G)) Sym(V(G)). 3 Graphs wth Gven Automorphsm Group Defnton 5 ([Fru39]). Gven a group Γ generated by elements S = {σ } I, the Cayley graph C(Γ, S) = (Γ, E) s the edge-labeled drected graph wth vertex set Γ and an edge x y wth label σ f σ S and y = σx. Theorem 3.1. Let Γ be a fnte group generated by S wth n = Γ. The Cayley graph C(Γ, S) has automorphsm group Γ. The labeled edges can be replaced wth smple undrected gadgets to form a graph C (Γ, S) of order O( Γ log S ) wth automorphsm group somorphc to Γ. For a whle, ths stood as the best upper bound on the sze of an undrected graph wth gven automorphsm group. Then, Sabduss presented n 1958 a complete characterzaton of the mnmum-order graphs wth a k-order cyclc automorphsm group for each k 2. Defnton 6. Let Γ be a fnte group. We defne the mnmum graph order α(γ) to be α(γ) = mn{n(g) : G = (V, E), Aut(G) Γ}, the mnmum order of a smple graph wth automorphsm group somorphc to Γ. Lemma 3.2 ([Sab59]). Let m 2 be an nteger. 2 f m = 2, 3m f m {3, 4, 5}, α(z m ) = 2m f m = p 3 7, p prme, t =1 α(z p e ) where m = t =1 pe for p 1,..., p t dstnct prmes. 4

5 Proof. We skp the mnmalty, but nstead focus on constructons that acheve these bounds. Note that Z 2 Aut(K 2 ). Moreover, f G 1,..., G t are graphs wth Aut(G ) Z e p and V(G ) = α(z e p ), for dstnct prmes p 1,..., p t, then ther unon G = t =1 G has automorphsm group Aut(G) t =1 Z p e Z p e pe t t Set m {3, 4, 5}. We construct G wth Aut(G) Z m and V(G) = 3m. Start wth the cycle C m. Subdvde each edge and nsert L(C m ) C m as the nduced subgraph of these new vertces. Now, the orgnal cycle has length 2m. Subdvde every other edge of ths cycle, and connect these m new vertces n an m-cycle. The vertces n the outer cycle of length 3m can be labeled x 1 y 1 z 1 x 2 y 2 z 2... x m y m z m n order. Ths gves that the vertces x have degree two, the y are n a cycle Y = y 1... y m and the z are n a cycle Z = z 1... z m. By the rotaton x x +1, we see that all ndces are adjusted by one, gvng the cyclc acton of Z m. Consder the nduced cycles Y and Z and an automorphsm π Aut(G). If π(y) = Y, then π nduces an acton on Y from D m, the dhedral group on m ponts. However, f π s a reflecton n D m, then π does not extend to G. Now, f m = p e 7, we can construct G from an m-cycle C = c 1... c m and and m-ndependent set X = x 1... x m. Consder all ndces modulo m. Note that the edges c c +1 are n C. Also add these edges: c x, c +2 x, c +3 x. Snce m 7, we see that each of, + 2, and + 3 defne unque vertces. Also, c +1 x s not an edge n G. Now, all vertces n X have degree three whle each n C has degree fve. Hence, X and C are stablzed by Aut(G). Moreover, snce C s stablzed and G[C] C m, Aut(G) s somorphc to a subgroup of D m, the dhedral group on m elements. Note that the constructon gves an automorphsm c c +1 and x x +1. So, Z m Aut(G) D m. The vertex x acts as an orentaton on the quadruple c c +1 c +2 c +3, snce the nduced subgraph of N[x ] s a trangle wth a leaf. Hence, we can determne an orentaton on each edge c c +1, as c +1 s the vertex between c (the leaf n N[x ]) and c +2 (one of the vertces n the trangle of N[x ]) and recover the ncreasng order of the ndces. Hence, Aut(G) Aut( Cm) Z m, where Cm s the drected cycle on m vertces. It wasn t untl 1974 when László Baba proved that those three cyclc groups were the only fnte groups that requred three vertces per element. All other fnte groups wth n elements are representable by a graph of order 2n. Theorem 3.3 ([Bab74]). If Γ s a fnte group not somorphc to Z 3, Z 4, or Z 5, then there exsts a graph G wth Aut(G) Γ and V(G) 2 Γ. Proof. If Γ s cyclc, we are done by Sabduss s theorem. If Γ V 4, we have V 4 Aut(K 4 e). Now, assume Γ > 6. Let S = {α 1,..., α t } be a mnmal generatng set of Γ. Create two graphs G 1 = (Γ, E 1 ), G 2 = (Γ, E 2 ). In G 1, for each element γ Γ and each {1,..., t 1}, place an edge between α γ and α +1 γ. Note that each vertex set {α 1 γ,..., α t γ} s a path n G 1. If there exsts an edge between α γ and α j γ wth j > + 1, ths contradcts mnmalty of S, snce there exsts γ Γ, l {1,..., t 1} so that α γ = α l γ, α j γ = α l+1 γ. Ths gves γ = α 1 l+1 α jγ and hence α = α l α 1 l+1 α j. In G 2, for each element γ Γ, place an edge between γ and α 1 γ. Both G s (s {1, 2}) are regular wth degree d s. We have d 2 = 2. If d 1 = d 2, then these graphs have the same degree.. 5

6 Defne G 3 by case: f d 1 d 2, then G 3 = G 2 ; f d 1 = d 2, then G 3 = G 2. Note that G 3 s regular wth degree d 3 d 1, snce f d 1 = d 2, then d 3 = n 1 d 2 = n 3 > 6 3 = 4 > d 2 = d 1. Defne G = (Γ {1, 3}, E) where E = E 1 (E 3 {3}) E, where Clam 3.1. Aut(G) Γ. E s = {{(γ, s), (δ, s)} : {γ, δ} E s }, E = {{(γ, 1), (γ, 3)} : γ Γ} {{(γ, 3), (α γ, 1)} : γ Γ, {1,..., t}}. Frst, note that Γ s somorphc to a subgroup of Aut(G). Gven δ Γ, π δ : V(G) V(G) s defned as π δ (γ, s) = (γδ, s) γ Γ, s {1, 3} Note that π δ defnes a bjecton on each edge set E 1, E 3, E as {(α γ, 1), (α +1 γ, 1)} π δ {(α γδ, 1), (α +1 γδ, 1)} (E 1 ) {(γ, 3), (α 1 γ, 3)} π δ {(γδ, 3), (α 1 γδ, 3)} (E 3 or E 3 ) {(γ, 1), (γ, 3)} π δ {(γδ, 1), (γδ, 3)} (E ) {(γ, 3), (α γ, 1)} π δ {(γδ, 3), (α γδ, 1)} (E ) It remans to show any permutaton n Aut(G) s represented by π δ for some δ Γ. Let γ Γ be any element. Defne the subgraph A γ be the nduced subgraph of G gven by (γ, 3), (γ, 1), (α 1 γ, 1),..., (α t γ, 1). As mentoned prevously, the vertces (α 1 γ, 1),..., (α t γ, 1) nduce a path n G. It s also true that there s no edge from (γ, 1) to (α γ, 1) for any {1,..., t}. If such an exsted, then there exsts an l {1,..., t 1} and γ Γ (γ γ) so that γ = α l γ, α γ = α l+1 γ. However, ths mples α = α l+1 α 1, whch contradcts mnmalty of S. l Hence, (γ, 1) s a leaf n A γ. Let π Aut(G) be a permutaton of V(G). Consder an element γ Γ and γ = π(γ). Snce π(a γ ) = A γ, and (γ, 1) s the only leaf n A γ, π(γ, 1) = π(γ, 1) snce (γ, 1) the only leaf n A γ. So, π can be consdered as a permutaton of Γ that also acts on G. Let π be such a permutaton gven by a non-trval automorphsm of G. Now, let γ be any element wth π(γ) γ and defne δ = γ 1 π(γ). Clam 3.2. For any element γ Γ, π(γ ) = γ δ. It s suffcent to prove that f π(γ) = γδ, then for all {1,..., t} has π(α γ) = α γδ. If ths s true, then for all γ Γ, the sequence of generators α j1 α jk = γ γ 1 gves γ = α j1 α jk γ and teraton on the number of generators n the rght-hand-sde product gves π(γ ) = γ δ. Snce the only vertex (α γ, 1) n A γ that has (α γ, 3) adjacent to (γ, 3) s (α 1 γ, 1). Hence, π(α 1 γ) = γδ. Moreover, the path (α 1 γ, 1)(α 2 γ, 1)... (α t γ, 1) n A γ s now embedded unquely nto π(a γ ) = A γδ as (α 1 γδ, 1)(α 2 γδ, 1)... (α t γδ, 1). Ths proves the clam. 6

7 4 Other Automorphsm Results Based on ths constructon of Baba, the worst-case order of a graph G wth automorphsm group Γ s O(n) where n = Γ. Unfortunately, we cannot hope for better asymptotcs than that (or much better constants, even), snce there s a very close lower bound for the alternatng group. Theorem 4.1 ([Le83]). If n 23, then the mnmum order of a graph wth automorphsm group somorphc to A n s at least 1 2 ( n n/2 ). Corollary 4.2. By Strlng s approxmaton, the above lower bound s approxmately 2n. 2πn The followng result s very recent and nterestng. We know the complexty of graph somorphsm s n NP but t s not known to be n conp. However, the planar case has a lnear-tme algorthm. Even more surprsng s the followng. Theorem 4.3 ([DLN + 09]). PLANARISOMORPHISM s n L. Ths theorem states there s a log-space algorthm to solve somorphsm for planar graphs. Ths result fnshed a seres of several papers n the past four years attemptng to tackle ths problem. It uses the fact that a 3-connected planar graph has a unque embeddng n the plane. Then, the graph G s decomposed nto 3-connected components, formng a tree-lke structure. An older algorthm of canonzng labeled trees s used to canonze G based on ths decomposton. References [Bab74] [Bol01] László Baba. On the mnmum order of graphs wth gven group. Canadan Mathematcal Bulletn, 17: , Béla Bollobás. Random graphs, volume 73 of Cambrdge Studes n Advanced Mathematcs. Cambrdge Unversty Press, Cambrdge, second edton, [DLN + 09] Samr Datta, Nutan Lmaye, Prajakta Nmbhorkar, Thomas Therauf, and Faban Wagner. Planar graph somorphsm s n log-space. In Proceedngs of the 24th Conference on Computatonal Complexty, [DP09] [Fru39] [Hof82] [Le83] [Sab59] Devdatt Dubhash and Alessandro Pancones. Concentraton of Measure for the Analyss of Randomzed Algorthms. Cambrdge Unversty Press, New York, NY, USA, R. Frucht. Herstellung von Graphen mt vorgegebener abstrakter Gruppe. Composto Math., 6: , Chrstoph M. Hoffmann. Group-Theoretc Algorthms and Graph Isomorphsm. Sprnger-Verlag, Martn W. Lebeck. On graphs whose full automorphsm group s an alternatng group or a fnte classcal group. Proceedngs of the London Mathematcal Socety, 3: , Gert Sabduss. On the mnmum order of graphs wth a gven automorphsm group. Monatsh. Math., 63: ,