Triangles in random graphs

Discrete Mathematics 289 (2004) 181 – 185 www.elsevier.com/locate/disc Note Triangles in random graphs Martin Loebl, Jiří Matoušek, Ondřej Pangrác Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Malostranské nám. 25, 118 00 Prague, Czech Republic1 Received 1 July 2003; received in revised form 15 July 2004; accepted 6 August 2004 Available online 11 November 2004 Abstract We show the number of triangles of Gn,1/2 is almost uniformly distributed among residue classes modulo q, where q is a prime number bounded by
(log n). This implies a consequence of a conjecture of Bollobás, Pebody and Riordan (that almost every random graph Gn,1/2 is uniquely determined by its Tutte polynomial): almost every pair of independently chosen random graphs Gn,1/2 has different Tutte polynomials. © 2004 Elsevier B.V. All rights reserved. Keywords: Number of triangles; Random graph; Tutte polynomial 1. Introduction The Tutte polynomial of a graph G = (V , E) is the bivariate polynomial
T (G; x, y) = (x − 1)r(E)−r(A) (y − 1)|A|−r(A) , A⊆E where r(A) = |V | − k(A) and k(A) denotes the number of components of the graph (V , A). It is clear that two isomorphic graphs have the same Tutte polynomial. On the other hand, there are examples of non-isomorphic graphs with the same Tutte polynomial (see [3,1]). E-mail address: [email protected] (M. Loebl), [email protected] (J. Matoušek), [email protected] (O. Pangrác). 1 Institute for Theoretical Computer Science is supported by Ministry of Education of the Czech Republic as project LN00A056. 0012-365X/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2004.08.008 182 M. Loebl et al. / Discrete Mathematics 289 (2004) 181 – 185 In this context, the question of how much information about the graph its Tutte polynomial contains is natural. A random graph Gn,1/2 , for n natural number, is a graph on an n element vertex set such that any pair of vertices forms an edge with probability 21 independently of other pairs. Let Gn,1/2 denote the corresponding probability space. Note that such a random graph is   dense; the expectation of the number of edges is 21 n2 and the expectation of the number   of triangles is 18 n3 . The following conjecture was presented in [1]. Conjecture 1.1. Almost every graph G ∈ Gn,1/2 is such that T (G; x, y) = T (G′ ; x, y) implies G ∼ = G′ . The main result of this paper stated below implies a weaker result. Theorem 1.2. There are numbers q0 and n0 and a function f ∈
(log n) such that for any n > n0 , any prime number q, q0 < q < f (n), and any choice of k ∈ {0, 1, . . . , q − 1} the probability that a random graph Gn,1/2 contains k triangles modulo q is q −1 (1 + o(1)). Since the number of triangles of any simple graph can be easily derived from its Tutte polynomial, we can state this easy consequence of Theorem 1.2. Corollary 1.3. The probability that two independently chosen random graphs Gn,1/2 have the same Tutte polynomial is of order O(1/ log n). Another open problem is to find or prove the existence of a family containing almost every graph such that the Tutte polynomial distinguishes any pair of graphs inside this family. Solving this problem affirmatively may give us better understanding and a hope for a positive answer to Conjecture 1.1. 2. Counting 2-paths Let G be a random graphfrom  Gn,1/2 with the vertex set V. Let V1 = {u1 , v1 , u2 , v2 ,. . ., u⌊ n ⌋ , v⌊ n ⌋ } be a set of 2 n4 vertices partitioned into pairs (ui , vi ), i = 1, 2, . . . , n4 . 4 4 i be an Let V2 be a subset of V − V1 and denote the size of V2 by l. For w ∈ V2 , let Xw i i indicator of the existence of both edges {ui , w} and {vi , w}. Then X = w∈V2 Xw counts the number of 2-paths connecting vertices ui , vi and having the middle vertex in V2 . Denote by pji (l) the probability that X i = j . Since edges from distinct vertices of V1 going to V2 are independent, the number of such 2-paths is independent for distinct pairs of vertices of V1 and hence we let pji (l) = pj (l). Also note that pj (l) depends only on l = |V2 |, not on the particular choice of V2 .  Assume q to be an arbitrary fixed integer 1
q
l. Denoted by si (l) = j ≡i(mod q) pj (l) the probability that the number of connecting 2-path with the middle vertex in V2 is equal to i modulo q. M. Loebl et al. / Discrete Mathematics 289 (2004) 181 – 185 183 Lemma 2.1. Let 1
q
|V − V1 |. There exists a number c = c(q) > 0 such that si (l)  c for i = 0, 1, . . . , q − 1 and l  q − 1. Proof. Note that the probability of a given 2-path is 41 . Using the recursion pi (l)= 41 pi−1 (l− 1) + 43 pi (l − 1) one can derive si (l) = 41 si−1 (l − 1) + 43 si (l − 1) (where s−1 (l) means sq−1 (l)). It is easy to observe that if si (l0 )  c for all possible i’s and some l0 and c, then also  q−1 si (l)  c for all l  l0 . Note that si (q − 1) = pi (q − 1) and clearly each pi (q − 1)  41 .  1 q−1 . By setting Hence for any l  q − 1 and any i = 1, 2, . . . , q − 1 we have si (l)  4  1 q−1 we complete the proof.  c(q) = 4 In the rest of the paper, we assume the pairs (u1 , v1 ), (u2 , v2 ), . . . , (u⌊n/4⌋ , v⌊n/4⌋ ) forming set V1 are fixed and V2 = V − V1 . Lemma 2.2. Let n be sufficiently large natural number. Then with probability at least 1 − 35 (log2 n)/n1/3 a graph G ∈ Gn,1/2 has, for any q
q0 = ⌊log2 n1/3 ⌋, the following property: there exists a (q − 1)-tuple (i1 , i2 , . . . , iq−1 ) such that the number of 2-paths connecting uik and vik is equal to k modulo q for any k = 1, 2, . . . , q − 1. Proof. By Lemma 2.1 the probability that the number of 2-paths with the middle vertex in V2 connecting ui and vi is equal to k modulo q is at least c(q) for any k = 0, 1, . . . , q − 1 (and hence at most 1 − (q − 1)c(q)). Moreover, these probabilities are independent for distinct pairs ui , vi and uj , vj . Denote the event that there is a pair ui , vi (among the pairs from V1 ) connected by exactly k 2-paths (counted modulo q) byYk (Yk = 1 if such apair exists,  Yk = 0 otherwise). Then the  probability P Y1 , Y2 , . . . , Yq−1 is equal to 1 − P ∃k : Yk . Assume the edges inside V1 are fixed. The probability that vertices ui , vi are connected by k 2-paths is si ′ (|V2 |), where i ′ is such that the number of 2-paths with the middle vertex inside V1 plus i ′ is equal to k   ⌊n/4⌋ modulo q. Hence, P Yk = i=1 (1 − si ′ (|V2 |))
(1 − c(q))⌊n/4⌋ . q−1 q−1 
 
 P Yk
(1 − c(q))⌊n/4⌋ = (q − 1)(1 − c(q))⌊n/4⌋ . P ∃k : Yk
k=1 k=1 Let q0 = ⌊log2 n1/3 ⌋. Since q
q0 and c(q)  c(q0 ) we can bound   (1 − c(q0 )2 )⌊n/4⌋ P ∃k : Yk
q0 (1 − c(q0 ))⌊n/4⌋ = q0 (1 + c(q0 ))⌊n/4⌋ q0 q0 n .

⌊n/4⌋ 1 + (1 + c(q0 )) 4 c(q0 ) 184 M. Loebl et al. / Discrete Mathematics 289 (2004) 181 – 185 From Lemma 2.1 we have c(q0 ) =   P ∃k : Yk
 1 q0 −1 4   1 log n1/3 4 = 2−2 log n log n1/3 5 log n log n1/3 n .
n −2/3 = 3 n1/3 n 1 + 4 n−2/3 5 1/3 = n−2/3 . Hence,  3. Proof of the Theorem Using the results of [2] we can state the following useful proposition. Proposition 3.1. Let q be a sufficiently large  prime number; then the number of subsets S ⊆ {1, 2, . . . , q − 1} such that the sum i∈S i ≡ k mod q is 2q−1 q −1 (1 + o(1)) for any k = 0, 1, . . . , q − 1. Combining the result of Lemma 2.2 with the previous proposition we can prove Theorem 1.2. Proof  of  Theorem 1.2. Let us consider a graph G ∈ Gn,1/2 with a fixed set V1 partitioned into n4 disjoint pairs of vertices as above. By Lemma 2.2 we can find with probability at least 1− 35 (log n)/n1/3 a (q −1)-tuple (i1 , i2 , . . . , iq−1 ) such that the number of connecting 2-paths between vertices uik and vik is equal to k modulo q for all k = 1, 2, . . . , q − 1. Choose such a (q − 1)-tuple lexicographically minimal. Let us define graphs GS for any S ⊆ {1, 2, . . . , q − 1} obtained from G by adding all edges {uik , vik } with k ∈ S (if they are not in G) and by deleting all edges {uik , vik } with k ∈ / S (if they are in G). The probability of all GS in the probability space Gn,1/2 is the same and the (q − 1)-tuple (i1 , i2 , . . . , iq−1 ) is lexicographically minimal with required property for any graph GS . Moreover, graphs G1 and G2 of Gn,1/2 are assigned the same (lexicographically minimal) (q − 1)-tuples (i1 , i2 , . . . , iq−1 ) and their edge sets differ only on the edges {uik , vik } (k = 1, 2, . . . , q−1) if and only if they are GS1 and GS2 for some G and S1 , S2 ⊆ {1, 2, . . . , q−1}. From now on, all computation will be done modulo q. Let t0 be the number of triangles of G∅ . Since the pairs uik , vik are disjoint, edges {uik , vik } do not affect the number of 2-paths between these pairs. Then the number of triangles of GS is t0 + k∈S k. By Proposition 3.1, if S is chosen uniformly at random then the probability that the number of triangles of GS is equal to k modulo q is q −1 (1 + o(1)). Up to a ( 53 (log n)/n1/3 )-fraction, we have partitioned the graphs of Gn,1/2 into classes of equal size such that in each class the number of triangles computed modulo q is almost uniformly distributed. Since (log n)/n1/3 tends to 0 for n going to infinity, we can conclude that the probability that Gn,1/2 contains exactly t triangles modulo q is q −1 (1 + o(1)) for any t = 0, 1, . . . , q − 1.  Acknowledgements The authors would like to thank D. Welsh for helpful discussions and M. Klazar for pointing out the paper of Erdős and Heilbronn. The authors would also like to thank the referees for suggesting several improvements. M. Loebl et al. / Discrete Mathematics 289 (2004) 181 – 185 185 References [1] B. Bollobás, L. Pebody, O. Riordan, Contraction-deletion invariants for graphs, J. Combin. Theory Ser. B 80 (2000) 320–345. [2] P. Erdős, H. Heilbronn, On the addition of residue classes mod p, Acta Arith. 9 (1964) 149–159. [3] H. Whitney, 2-isomorphic graphs, Amer. J. Math. 55 (1933) 245–254.