Discrete Mathematics. On the number of graphs with a given endomorphism monoid

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1 Discrete Mathematics Contents lists available at ScienceDirect Discrete Mathematics journal homepage: On the number o graphs with a given endomorphism monoid Václav Koubek a,, Vojtěch Rödl b a Department o Theoretical Computer Science and Mathematical Logic, The Faculty o Mathematics and Physics, Malostranské nám. 5, 8 00 Praha, Czech Republic b Department o Mathematics, Emory University, Atlanta, USA a r t i c l e i n o a b s t r a c t Article history: Received March 009 Accepted March 009 Available online 4 April 009 Dedicated to Proessor Jiří Sichler on his 65th birthday Keywords: Endomorphism monoid o graph Rigid graph For a given inite monoid M, let ς M n be the number o graphs on n vertices with endomorphism monoid isomorphic to M. For any nontrivial monoid M we prove that n +odmn ς M n n +ocmn where cm and dm are constants depending only on M with.83 cm dm 3 M M 3. For every k there exists a monoid M o size k with dm 3, on the other hand i a group o unity o M has a size k > then cm log k +. log log k 009 Elsevier B.V. All rights reserved.. Introduction An undirected graph is a pair G = X, E where VG = X is a set the set o vertices o G and EG = E is a set o two-element subsets o X the set o edges o G. Given two undirected graphs G = V, E and G = V, E, a mapping : V V is called a graph homomorphism rom G into G i { u, v} E or all {u, v} E. I G = G = G then a graph homomorphism : VG VG is called an endomorphism o G. Let EndG denote the set o all endomorphisms o G and AutG consist o all automorphisms o G an automorphism is a bijective endomorphism such that is also an endomorphism. Note that the identity mapping belongs to both EndG and AutG and both sets EndG and AutG are closed under composition. Thus EndG is a transormation monoid and AutG is a permutation group or any graph G and i G is inite then also EndG and AutG are inite. An old result o Hedrlín and Pultr [,3] states that the opposite implication is valid as well: For any inite monoid M there exists a inite graph G such that EndG is isomorphic to M. This act inspired an investigation o the endomorphism monoids o graphs and also o other structures. A survey o these results can be ound in a monograph o Pultr and Trnková [7]. The monoid, consisting o the unity element only, will play an important role. I an endomorphism monoid o a graph G is isomorphic to the only identity mapping is an endomorphism o G then we say that G is rigid. Graphs G and G or which there exists no homomorphism between them are called mutually rigid. The irst rigid graph was constructed by Hedrlín and Pultr [3] see also [7]. It is also known that a rigid graph exists on every set with at least eight vertices, see [7]. Hedrlín Sichler Theorem [4] or [7] states that or every inite monoid M there exist ininitely many inite pairwise mutually rigid graphs with endomorphism monoid isomorphic to M. On the other hand, a result o [6] states that almost every inite graph is rigid. The aim o this note is to continue this research. For a inite monoid M and a natural number n we will consider the ollowing two parameters. The authors grateully acknowledge the support o M a project o the Czech Ministry Education. The irst author grateully acknowledges the support o the Grant No ET o Inormation Society and the second author grateully acknowledges the support by NSF Grants DMS and DMS Corresponding author. addresses: koubek@ktiml.m.cuni.cz, koubek@ksi.ms.m.cuni.cz V. Koubek, rodl@mathcs.emory.edu V. Rödl X/$ see ront matter 009 Elsevier B.V. All rights reserved. doi:0.06/j.disc

2 V. Koubek, V. Rödl / Discrete Mathematics ς M n the number o labeled graphs G on a ixed n-element set such that EndG is isomorphic to M, ρ M n the largest integer k such that on a ixed n-element set there exist at least k pairwise mutually rigid graphs each with endomorphism monoid isomorphic to M. For simplicity, we recall that a unction = o i lim n n = 0. A logarithm in this paper is always a logarithm o base. The ollowing was proved in [6]. Theorem [6]. For the singleton monoid ς n = + o n, ρ n = + o n! n n. The aim o this note is to extend Theorem and to improve Theorem.8 rom [6] where the weaker lower bound with the constant dm replaced by + ɛ log n was given. Theorem. For every inite monoid M there exist constants cm and dm depending only on M with.83 5 log 9 cm dm 3 M M 3 such that n +odmn ς M n n +ocmn, n dm + o ρ M n + o n! n! n dm n cm n cm. Remark. We conjecture that or every monoid M there exists a constant am such that ς M n = n +oamn. In n other words, lim n log exists and equal to am. n ς M n While Theorem states that cm 5 log 9 or every monoid M, our next result states that, perhaps surprisingly, even i the size o M is large, this cannot be essentially improved. Theorem 3. For every k > there exists a monoid M o size k with dm 3. Thus or every natural number k > there exists a monoid M o size k with 5 log 9 cm dm 3. For a inite monoid M with the unity element e, let GrM consist o all elements x o M such that x t = e or a suitable positive integer t. It is well known that GrM is a maximal subgroup o M containing the unity element e. I G is a graph such that EndG is isomorphic to M then AutG is isomorphic to GrM thus AutG = GrEndG. Our next result shows a dependency between cm and GrM. The monoid M used in the proo o Theorem 3 is such that GrM is a singleton group. In contrast to this result we prove the ollowing. Theorem 4. For every inite group G o size k >, the number o pairwise distinct graphs G on an n-element set such that AutG is isomorphic to G is at most n log k n log log k + o. log GrM Thus or every inite monoid M such that GrM is not a singleton, we have cm log log GrM + o.. Proo o lower bounds rom Theorem To establish the lower bound we need to construct many graphs with a given endomorphism monoid. For this purpose we exploit the ollowing class o graphs. Given δ > 0 we call a graph H on m vertices δ-restricted i i or every x VH δ m deg H x + δ m, ii the size o every clique and every independent set in H does not exceed + δ log m.

3 378 V. Koubek, V. Rödl / Discrete Mathematics Fig.. Construction o G = GG 0, H, C, φ. The irst statement o the act below is well known see e.g. [], the second statement can be observed along the same lines. Fact 5. For each δ > 0, the number o δ-restricted labeled graphs with m vertices is + o m, and the number o δ-restricted labeled graphs with m vertices and m + o m! m m edges is Combining Theorem and Fact 5 we iner that the quantities in Fact 5 remain unchanged even i we restrict to rigid graphs. Fact 6. For each δ > 0, the number o δ-restricted rigid graphs with m vertices is + o m, and the number o mutually rigid δ-restricted graphs with m vertices and m + o m! m m edges is Our strategy in proving the lower bound is to ix one graph G 0 on k vertices with EndG 0 = M and combine it with many rigid δ-restricted graphs H on m vertices to construct many graphs G with EndG = M on n < m + k vertices. For a graph G 0 = V 0, E 0, let us deine a partition o vertices into moving and solitary sets V 0 = M S, where M = MovG 0 = {x V EndG, x x or x {x}} and S = SolG 0 = V 0 \ M. The proo o the lower bound is based on the ollowing construction. Let G 0 = V 0, E 0 and H = W, F be two graphs, let C W be a maximal clique o H and let φ be an injective mapping rom S = SolG 0 into W such that Imφ C =. We will identiy any vertex x S with the vertex φx W. We are going to deine a graph c. Fig. GG 0, H, C, φ = G = V, E. A vertex set V is the union o MovG 0 and W. The edge set E is the union o F, E 0 \ between C and M = MovG 0. More ormally, V = MovG 0 W = M W and E = F {{u, v} E 0 {u, v} MovG 0 } {{u, φv} {u, v} E 0, u MovG 0, v SolG 0 } {{u, c} u MovG 0, c C}. S and the complete bipartite graph As above, we assume that the graphs G 0 = V 0, E 0, H = W, F a maximal clique C VH and an injective mapping φ : SolG 0 W are given with k 0 = V 0, M 0 W = and Imφ C =. First we prove two auxiliary lemmas. Claim 7. Let H and H be 4 -restricted graphs such that VH = VH = m. I m > 50 log m and k 0 5/ log m then every homomorphism : H G = GG 0, H, C, φ satisies Im W.

4 V. Koubek, V. Rödl / Discrete Mathematics Proo. Suppose that there is w VH with w = v M. Since deg H w m 4 or every w VH and deg G v < k log m < 5 log m then there is a neighbour u o v in G such that u 0 m log m However, u must be an independent set in -restricted graph 4 H, and thereore, + log m m 4 0 log m which is a contradiction. Claim 8. I H is 4 -restricted graph on m vertices, where m > 50 log m and k 0 5 log m, and G = GG 0, H, C, φ then I H is rigid, a mapping : V V is an endomorphism o G i and only i there exists g EndG 0 such that v = gv or all v M and w = w or all w M. I H is 4 -restricted graph with VH = m such that H and H are mutually rigid then there is no homomorphism rom H to G. Proo. Let EndG and w W. By Claim 7, w W and because H is rigid, w = w. For every v MovG = M, C {v} is a clique o G. Consequently, since c = c or every c C, we iner that v W. For otherwise, C would not be a maximal clique. Consequently, M M. Consider a mapping g : V 0 V 0 such that gv = v or all v M and gv = v or all v S = SolG 0. Since v = v or all v Imφ and since EndG, we conclude that g EndG 0. Conversely, i g EndG 0 then we deine : V V such that w = w or all w W and v = gv or all v M. Since g is an endomorphism o G 0 and since w = w or all w W, one may easily veriy that EndG. Consequently, is proved. To prove assume now that : H G is a homomorphism. By Claim 7, Im W and the range restriction o on W is a homomorphism rom H into H this is a contradiction and is proved. Now we are ready to inish the proo o the lower bounds rom Theorem. Let M be a inite monoid and let G 0 = V 0, E 0 be a graph such that EndG 0 = M and or which MovG 0 is as small as possible. We set d = MovG 0 V 0 by a result o [6], see Construction.4, d + +8 M + M 3 M + M 3. As beore we write V0 = M S or a decomposition into moving and solitary parts o V 0. For an integer n, set m = n d. Let {H i, C i, φ i i I} be a amily such that or all i I i H i is a -restricted rigid graph, 4 ii W = VH i, W V 0 =, W = m, iii C i is a maximal clique o H i, and iv φ i : S W is an injective mapping such that C i Imφ i. Since Theorem is an asymptotic statement, we may assume that m is suiciently large, more precisely m > 50 log m and V 0 = k 0 5 log m and hence the assumptions o Claim 8 are satisied. Consequently, GG 0, H i, C i, φ i = EndG 0 or each i I and hence by Fact 6 ς M n o m n d = o = n o+dn. Thus there exists dm d + = 3 M M 3 satisying the upper bound rom Theorem. Similarly, Fact 6 implies the lower bound or ρ M n. 3. Proo o upper bounds rom Theorems and 4 In the proo o Theorem 4 we will use the ollowing claim which we prove ater the proo o Theorem 4. Claim 9. The number o graphs G = X, E with X = n such that there exist an EndG and a set Z X such that Z = l, Z Z = and is injective on Z is at most +n n log n log 3 l. Proo o Theorem 4. For a permutation : X X, let UNSO denote the union o all nonsingleton orbits o. Similarly, or a permutation group P on a set X, let UNSOP be the union o all nonsingleton orbits o P. Fix an abstract group Γ o size k >. We shall estimate the number o graphs G on a set X such that AutG is isomorphic to Γ. Since i any permutation : X X with UNSO 3l satisies the assumptions o Claim 9 or a set Z X with Z = l and ii any group Γ o size k has a set o generators o size at most log k.

5 380 V. Koubek, V. Rödl / Discrete Mathematics Claim 9 implies that the number o graphs G = X, E such that UNSOAutG 3n 0.5 log k and AutG is isomorphic to Γ is at most +n n n log n log n log 3 on.0 which is much smaller than the upper bound o Theorem 4. Thus we can assume that UNSOAutG < 3n 0.5 log k. To count the number o graphs G = X, E such that AutG is isomorphic to Γ and UNSOAutG < 3n 0.5 log k consider a amily S o pairwise disjoint subsets o X such that A > or any A S and A S A = m < 3n0.5 log k. Since AutG acts on m elements we conclude that m! k and hence m log k. First we estimate the number a log log k S o graphs G = X, E such that the set o nonsingleton orbits o AutG is S and AutG is isomorphic to Γ. Consider z X such that z = z or all AutG. Then or x and y rom the same orbit o AutG we have {x, z} E i and only i {y, z} E. Setting S = q we have a S n m m n mq n n n mm q. Since m q we conclude that m q log k and thus because k is ixed log log k n mm q n 3n 0.5 log k log k log log k = o n log k log log k. Hence we conclude that a S n log k n o log log k n = n log k n log log k + o. Having established the upper bound or a S we will now estimate the number o choices o orbits o AutG. This is clearly bounded above by we set m 0 = 3n 0.5 log k { n q m m m m0, q m } { n m m! qm m m 0, q m } { eq m n m m m0, q m } m { e m0 n m m0, q m } m 0 e m0 e om0 n = n Hence the number o pairwise distinct graphs G = X, E such that AutG is isomorphic to Γ and UNSOAutG < 3n 0.5 log k is less than a S n log k n log log k ++o e +om0 n n log k n log log k ++o S S = n log k n log log k ++o. Summing both estimates depending on the cardinality o the union o nonsingleton orbits o AutG completes the proo o Theorem 4. Proo o Claim 9. Let X = n. Fix a mapping : X X such that there exists a set Z X with Z = l, Z Z = and is injective on Z. For any graph G = X, E with EndG, i x, y Z are distinct then { x, y} E whenever {x, y} E. Hence or each x, y Z, x y, E {{x, y}, { x, y}} is one o the ollowing three sets, {{ x, y}} and {{x, y}, { x, y}}. Consequently or such there exist at most n l l 3 = n 3 4 l graphs G = X, E with EndG. Summing up over all such mappings yields at most l n n n 3 +n n log n log 3 l 4 graphs G satisying the assumptions o Claim 9.

6 V. Koubek, V. Rödl / Discrete Mathematics Remark. In act we proved the stronger estimate. For a group Γ let αγ = minm G q G where the minimum is taken over all permutation groups G isomorphic to Γ and m G = UNSOG and q G is the number o nonsingleton orbits o G. Then the number o pairwise distinct graphs G = X, E such that AutG is isomorphic to Γ is at most n αγ + on. Proo o the upper bound in Theorem. Let M be a nonsingleton inite monoid with unity element e. I GrM = {e}, we can guarantee the existence o a nonidentical idempotent in M. Indeed i GrM = {e} then, by the assumption on M, the set M\GrM is nonempty. Choose x M\GrM. For some k, t N we have x k = x k+t. Choose i N such that i+k 0 mod t. Then x i+k = x i+k+i+k = x i+k+st = x i+k or some s. The number o graphs having a nonidentical automorphism is bounded by Theorem 4 and Remark ater the proo o log k Theorem 4. By a routine calculation we obtain that + 5 log 9 or all k 3. I Γ is a group o cardinality then log log k αγ = and Remark gives that the number o pairwise distinct graphs G = X, E such that AutG is isomorphic to Γ is at most n 3+on which is smaller than the upper bound o Theorem. Consequently, it is suicient to veriy this upper bound in the case that there exists an idempotent nonidentical EndG. We prove that the number o pairwise distinct graphs G on a set X such that there exists a nonidentical idempotent endomorphism is at most n 5+o log 9n and hence the upper bound rom Theorem ollows. The proo uses the ollowing lemma which will prove ater the proo o Theorem. For an idempotent mapping : X X, let us deine Fix = {x X x = {x}} and St = Im \ Fix = {x X y X, y x, y = x} see Fig.. For x St, let us denote a x = x and also set a = {a x x St } and b = St. Lemma 0. For every nonidentical idempotent mapping : X X where X = n, the number o pairwise distinct graphs G on a set X such that EndG is at most 9 ζ X, = 6 b a a b a + a n a n. The proo is divided into three cases. In each case a numerical constraint is made on a and b, respectively, and the number o graphs G on a set X admitting an idempotent endomorphism satisying the corresponding constraint is counted. The bound 3 is obtained as a sum o terms derived in each o the cases. We use the ormula rom Lemma 0 and depending on considered case will neglect some o its terms which are smaller than one as we ind appropriate. We will see that only the contribution coming rom Case is essential or the bound 3. Case : a < n Since the number o subsets o an n-element set o size a is at most n n0.95 we conclude that there are at most n n0.95 n 0.95 n 0.95 =.95n0.95 log n +on0.96 idempotent mappings on X with a < n Consequently, by Lemma 0, the number o pairwise distinct graphs G = X, E such that there exists an idempotent EndG with a < n 0.95 is at most ζ X, = = = 9 b 6 a + a a a n a n b a + a n a log + a n a n n n 3 +o log + a n n a n n 5 n +o log 3 +on0.96 n 5+o log 9n = n 5+o log 9n we note that on <, + the unction 3 log + x attains minimum or x = and 3 log + = 5 log 3. 3

7 38 V. Koubek, V. Rödl / Discrete Mathematics Fig.. Fix point set Fix and stable set St o an idempotent. Case : b n 0.6. ζ X, = 9 6 b n n n n. log 4 3 n = n +on.. n. n 0.6 log 9 6 n Case 3: n 0.6 b and a n In this case ζ X, = a a b n n log n 0.5n n.9 n 0.6 n +on.. Proo o Lemma 0. Fix an idempotent mapping : X X with a >. We estimate the number o graphs G = X, E with EndG. Observe that {u, v} E implies {x, y} E whenever x, y Im with x y and u x, v y. On the other hand {u, v} E i there exists x Im with u, v x. Hence the number o pairwise distinct graphs G on X such that EndG is at most Since i I a i { a x a y + x, y St, x y} { ax + n a n a x St }. 4 i I a i I and a xa y a xa y i a x, a y we deduce that { a x a y + x, y St, x y} { a xa y x, y St, x y b Further observe that i b and c are integers with < c b then a c + b + = b+c + b + c + b+c + b + = b+c + b + b+c + } a b. 5 because b b+c. Hence by an easy induction we obtain that { a x + n a x St } a + n a. 6 Combining 4 6 and n a n a = + an a + we obtain that the number o graphs G on the set X with EndG is at most b 9 a a b a + n a n a 6 b 9 = a a b a n a + 6 a n.

8 V. Koubek, V. Rödl / Discrete Mathematics Proo o Theorem 3 To prove Theorem 3 consider a class RZ o right-zero monoids. Let X be a set with a distinguished point e X and let be a binary operation such that { y i y e x y = x i y = e or x, y X. Then X with the operation and the element e is a monoid such that e is the unity element. This monoid is called a right-zero monoid. We prove that, or every inite monoid M RZ, the number o pairwise distinct graphs on an n-element set such that its endomorphism monoid is isomorphic to M is at least n o3n. Theorem 3 ollows because or every natural number m there exists a right-zero monoid o size m. The proo o this act is based on the ollowing generalization o Theorem which we prove ater the proo o Theorem 3. Fact. For every integer k, the number o pairwise distinct rigid graphs G = Y, E on an n-element set Y such that or every k-element set Z Y we have 0.9n k {y Y z Z, {y, z} E}.n k ; or every k + -element Z Y we have is + o n. 0.9n.n {y Y z Z, {y, z} E} k+, k+ Proo o Theorem 3. Let M RZ be a right-zero monoid o size k >. Set m = n, or m > k, let G = Y, E be a rigid graph on an m-element set Y such that or every k-element set Z Y we have 0.9m k {y Y z Z, {y, z} E}.m k ; or every k + -element set Z Y we have 0.9m.m {y Y z Z, {y, z} E} k+. k+ Choose a vertex u Y and a k-element set Z = {z, z,..., z k } Y. Let N Z be a neighbourhood o Z, i.e. N Z = {y Y z Z, {z, y} E}. Let H = X, F be a graph such that X = Y {u} and F = E {{y, u} y N Z }. We prove that EndH is isomorphic to M. For every i =,,..., k let i : X X be a mapping such that i y = y or all y Y and u = z i. It is a routine to veriy that i is an endomorphism o H or all i =,,..., k. Conversely, let be an endomorphism o H. Then i is an endomorphism o H with Im i Im i Y. Hence the domain-range restriction g i o i to Y is an endomorphism o G. By the assumption on G, g i is the identity mapping or all i =,,..., k. Since or every y Y there exists i =,,..., k with y z i, then i y = {y}, and we conclude that y = i y = g i y = y or all y Y. Now show that u Z {u}. Indeed, i u = y Z {u} then {y, x} E or all x N Z and this contradicts the assumption on G because N Z 0.9n >.n {x X z Z {y}, {x, z} E}. k k+ Summarizing these acts we iner that EndH = { i i =,,..., k} {ι} where ι is the identity mapping o X. Since i j = j or all i, j =,,..., k we conclude that EndH is isomorphic to M. Let us denote X = n and apply Fact then the number o pairwise distinct graphs on X such that its endomorphism monoid is isomorphic to M is at least + o n = n +o3n. The proo o Theorem 3 is complete. The proo o Fact exploits the ollowing Cherno bound. Lemma [5]. For all positive integers m and k and or reals p and q with p + q = and or positive reals ε < 3 we have m p t q m t e ε 3 mp t where the sum is taken over all integers t such that t mp εmp.

9 384 V. Koubek, V. Rödl / Discrete Mathematics Proo o Fact. Consider a random graph G G n, with n vertices where the edges are chosen independently each with the probability. Fix a k-element set Z [n]. The expected size o its joint neighbourhood N G n, Cherno inequality with m = n k and p = k the probability that N G n, Z n k k n k 0 k Z is n k k. Thus using holds is bounded above by e n k 00 k. Let B k be the event that there exists a k-element set Z so that 7 holds. Then the probability n PB k e n k 00 k = o. k Let B k+ be the event regarding k + -element sets deined analogously. Then the probability PB k B k+ = o. Consequently, all but o n graphs G Gn, satisy N G n, or all Z N G n, Z n j j n j 0 j and j = k or j = k +. Since 8 implies n j Z n j 0 n j or suiciently large n, and since, by Fact 5, almost all graphs are rigid, Fact ollows. Reerences [] P. Erdös, J.H. Spencer, Probabilistic Methods in Combinatorics, Akadémiai Kiadó, Budapest, 974. [] Z. Hedrlín, A. Pultr, Relations graphs with initely generated semigroups, Monatsh. Math [3] Z. Hedrlín, A. Pultr, Symmetric relations undirected graphs with given semigroups, Monatsh. Math [4] Z. Hedrlín, J. Sichler, Any boundable binding category contains a proper class o mutually disjoint copies o itsel, Algebra Universalis [5] S. Janson, T. Luczak, A. Rucinski, Random Graphs John Wiley &SONS, INC, A Wiley Interscience Publication New York. [6] V. Koubek, V. Rödl, On the minimum order o graphs with given semigroup, J. Combin. Theory Ser. B [7] A. Pultr, V. Trnková, Combinatorial, Algebraic and Topological Representations o Groups, Semigroups and Categories, North-Holland, Amsterdam,