Counting trees in graphs

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1 Coutig trees i graphs Dhruv Mubayi Jacques Verstraëte August 4, 206 Abstract Erdős a Simoovits proved that the umber of paths of legth t i a - vertex graph of average degree d is at least ( δ)(d ) (d t + ), where δ = (log d) /2+o() as d. I this paper, we stregthe a geeralize this result as follows. Let T be a tree with t edges. We prove that for ay -vertex graph G of average degree d t, the umber of labelled copies of T i G is at least ( ε)(d ) (d t + ) where ε = (4t) 5 /d 2. This bou is tight except for the term ε, as show by a disjoit uio of cliques. Our proof is obtaied by first showig a lower bou that is a covex fuctio of the degree sequece of G, a this aswers a questio of Dellamoica et. al. Itroductio If H a G are graphs, let N H (G) deote the umber of labelled copies of H i G. More precisely, N H (G) is the umber of ijectios φ : V (H) V (G) such that φ(u)φ(v) E(G) for every edge uv of H. Let N G (v) deote the eighborhood of vertex v i a graph G a let d G (v) = N G (v). Let (a) b = a(a )(a 2)... (a b + ) whe a b a a is real a b is a iteger, a let (a) 0 =. Let T be a t-edge tree a G = (V, E) be a graph with miimum degree at least t. I this paper, we obtai lower bous o N T (G). This is a basic questio i combiatorics, for example, the simple lower bou (d(v)) t i the case whe T is a star is the mai iequality eeded for a variety of fuametal problems i extremal graph theory. Coutig paths a walks i graphs has umerous applicatios, such as fiig bous o the spectral radius of the graph a the eergy of a graph (see Täubig a Weihma [6] a the refereces therei), as well as Departmet of Mathematics, Statistics, a Computer Sciece, Uiversity of Illiois, Chicago, IL, USA. Research partially supported by NSF grat DMS mubayi@uic.edu Departmet of Mathematics, UCSD. jacques@ucsd.edu the electroic joural of combiatorics 22 (205), #P00

2 i staard graph theoretic reductios of combiatorial problems such as coutig words over alphabets with forbidde patters. Blakley a Roy [2] used elemetary liear algebraic methods to show that if A is ay symmetric poitwise o-egative matrix a v is ay poitwise o-egative uit vector, the for ay t N, A t v, v Av, v t. If t is eve, this is a simple cosequece of the fact that A is diagoalizable a Jese s Iequality applied to a covex combiatio of the t-th powers of the eigevalues of A. Whe A is the adjacecy matrix of a -vertex graph of average degree d, this shows that the umber of walks of legth t is at least t. There are ow a umber of proofs of this result, for istace aother approach to coutig walks was used by Sidoreko, usig a aalytic method a the tesor power trick [5], as well as geeralizatios to various other iequalities ivolvig walks (see [6] a the refereces therei). Alo, Hoory a Liial [] showed that the umber of o-backtrackig walks i a - vertex graph G of average degree d a miimum degree at least two is at least (d ) t, which is agai tight for d-regular graphs. A key fact i their proof is that if oe cosiders a raom walk of legth t o the graph, startig with a uiformly raomly chose edge, the the distributio of every edge of the walk is also uiform. The method of Alo, Hoory a Liial [] was geeralized i Dellamoica et. al. [3] to show that the umber of homomorphisms of a t-edge tree T i G is at least d(v)(t )d(v)/. By Jese s Iequality, this is at least t whe G has o isolated vertices. I Problem 4 of [3], the questio of fiig similar lower bous for N T (G) whe T is a tree is raised. I the case T is a path with three edges, it was show i [3] that N T (G) (d 2)2d(v)/d whe G has miimum degree at least three. If P is a t-edge path, a G is a d-regular graph, the clearly N P (G) (d) t with equality for d t if a oly if every compoet of G is a clique. Erdős a Simoovits [4] showed that if G is ay graph of average degree d, the N P (G) (d o(d)) t as d. I this paper, we aswer Problem 4 of [3], exte the result i [4] to arbitrary trees a stregthe their lower bou to (d O(/d)) t as d : Theorem. If T is ay t-edge tree a G is a -vertex graph of average degree d t, a ε d = (4t) 5 /d 2, the N T (G) ( ε d )(d) t. This result is best possible up to the factor ε d i view of the graph comprisig /(d + ) cliques of order d + t + whe d + is a positive iteger dividig, sice these graphs have exactly (d) t copies of T. Note that sice ε = O(/d 2 ), Theorem implies N T (G) (d O(/d)) t as d. Sice we do ot have a example of a graph G with vertices a average degree d for which N T (G) < (d) t, we propose the followig cojecture: Cojecture 2. For every t >, there exists d 0 = d 0 (t) such that for ay tree T with t edges a ay -vertex graph G with average degree d d 0, N T (G) (d) t. the electroic joural of combiatorics 22 (205), #P00 2

3 Oce more the example of disjoit complete graphs of order d+ shows that this cojecture if true is best possible. Perhaps it is eve true that if d is a iteger, the this is the oly example where equality holds. This paper is orgaized as follows. I the ext sectio, we determie a lower bou o N T (G) for a t-edge tree T, as a fuctio of the degree sequece of G. This aswers Problem 4 of [3]. The proof of this result extes the method of [] by carefully cotrollig the distributio of the edges i a raom embeddig of a tree i a graph of average degree d. The fuctio of degrees is almost covex, a i Sectio 3 we give a suitable approximate form of Jese s Iequality, which is used i Sectio 4 to derive Theorem. 2 Degree sequeces a coutig trees As a steppig stoe to Theorem, we are goig to prove the followig theorem, which gives a lower bou for N T (G) i terms of the degree sequece of G, a addresses Problem 4 i [3]: Theorem 3. Let T be a t-edge tree, a let G = (V, E) be a -vertex graph with average degree d a miimum degree k > t >. The where for 3 i t +, a for 3 i t +, N T (G) t+ (d(v) i + 2) d(v) i+2 i=3 (d i+2) θ i(v) (d(v) i + 2)θ i (v) = d i + 2 (2) t ) t θi (v) k () ) t (3) k t Proof. Let Ω be the set of all labelled copies of T i the graph G. Fix a breadth-first search orderig x = (x, x 2,..., x t+ ) of the vertices of T, startig with a leaf x of T. For 2 i t +, let a(i) be the uique h < i such that x h is the paret of x i i the orderig x. Note that a(2) = a a(3) = 2. Let v v 2 be a raomly a uiformly chose (orieted) edge of G a defie φ(x ) = v a φ(x 2 ) = v 2. Havig defied φ(x j ) = v j for all j < i, let x i be mapped to a uiformly chose vertex of N + (v a(i) ) := N(v a(i) )\{v, v 2,..., v i }. Oe ca always defie φ i this maer sice d(v a(i) ) k t + i a therefore N + (v a(i) ) N(v a(i) ) {v, v 2,..., v i }\{v a(i) } d(v a(i) ) i + 2 > 0. the electroic joural of combiatorics 22 (205), #P00 3

4 Here we oted that v a(i) {v, v 2,..., v i }. The φ gives a probability measure P o the sample space Ω. Let v deote a vector (v, v 2,..., v t+ ) of vertices of G. We write φ( x) = v to deote the evet that φ(x i ) = v i for all i [t + ]. The By the iequality of arithmetic a geometric meas: P(φ( x) = v) = t+ N i=3 + (v a(i) ). (4) N T (G) = v Ω P(φ( x) = v) v Ω P(φ( x) = v) P(φ( x) = v) P(φ( x)= v) = () P(φ( x)= v) t+ N + (v a(i) ) P(φ( x)= v) v Ω v Ω i=3 = t+ N + (v a(i) ) P(φ( x)= v). v Ω i=3 Let Ω iv = { v Ω : v a(i) = v}. Sice N + (v a(i) ) d(v a(i) ) i + 2 for 3 i t +, N T (G) t+ N + (v a(i) ) P(φ( x)= v) v Ω i=3 t+ = N + (v a(i) ) P(φ( x)= v) i=3 v Ω iv t+ (d G (v) i + 2) P(φ( x)= v) i=3 v Ω iv t+ = (d G (v) i + 2) v Ω P(φ( x)= v) iv i=3 t+ = (d G (v) i + 2) P(φ(xa(i))=v). (5) i=3 I the ext two claims, we fix v V. For i [t + ], defie g i (v) = P(φ(x i ) = v). We observe that g (v) = g 2 (v) = d G (v)/d. For 3 i t +, we determie upper a lower bous o g i (v) relative to d G (v)/d. We write d(v) istead of d G (v) i what follows. the electroic joural of combiatorics 22 (205), #P00 4

5 To bou the quatities g i (v), let T i be the subtree of T cosistig of the vertices x, x 2,..., x i i order. Now cosider a breadth-first search orderig y, y 2,..., y i of T i such that y = x i a y i = x. For 2 j i, let b(j) be the uique h < j such that y h is the paret of y j i the orderig y, y 2,..., y i a let N + (u b(j) ) = N G (u b(j) )\{u, u 2,..., u j }. As a example, if T i is a path, the b(j) = j a N + (u j ) = N G (u j )\{u, u 2,..., u j }. Claim. Let 3 i t + a let v = u b(2) = u. The u 2 N + (u b(2) ) u i N + (u b(i) ) j=3 d(u b(j) ) g i(v) u 2 N + (u b(2) ) u i N + (u b(i) ) j=3 d(u b(j) ) t (6) Proof of Claim. The umber of choices of a embeddig of T i with x i mappig to v is. (7) u 2 N + (u b(2) ) u 3 N + (u b(3) ) u i N + (u b(i) ) Here we are coutig images of T i accordig to the breadth-first search orderig y, y 2,..., y i of T i with y = x i a y i = x, so that the image of y j is u j for j [i]. At the stage where y j is to be embedded, we have to select a vertex u j N G (u b(j) ) such that u j is ot equal to ay of v = u, u 2,..., u j, which have already bee embedded. By defiitio of N + (u b(j) ), this is equivalet to u j N + (u b(j) ). Suppose we fix such a embeddig with vertices u, u 2,..., u i. Let v, v 2,..., v i be the re-orderig of u, u 2,..., u i such that x j is mapped to v j for j [i]. This fixed embeddig has probability P(φ( x) = v) = Sice d(v a(j) ) t N + (v a(j) ) d(v a(j) ), j=3 d(v a(j) ) j=3 P(φ( x) = v) N + (v a(j) ). (8) j=3 d(v a(j) ) t. (9) We ow prove {v a(j) : 3 j i} a {u b(j) : 3 j i} are equal as multisets. For example, if T i is a path, the these multisets are sets, a {v a(j) : 3 j i} = {v 2, v 3,..., v i } = {u 2, u 3,..., u i } sice u j = v i j+ for j [i]. For 2 l i, the umber of times u l appears i the multiset {u b(j) : 3 j i} is the umber of j with 3 j i such that b(j) = l. I other words, it is the umber of times u l is a paret of some vertex u j i the orderig u,..., u i. This is precisely d T (y l ). Similarly, the umber of times u l {v 2, v 3,..., v i } appears i the multiset {v a(j) : 3 j i} is the the electroic joural of combiatorics 22 (205), #P00 5

6 umber of times u l is a paret of some vertex v j i the orderig v,..., v i, a this is agai d T (y l ). We coclude {v a(j) : 3 j i} a {u b(j) : 3 j i} are equal as multisets. Therefore by (9), j=3 d(u b(j) ) P(φ( x) = v) j=3 d(u b(j) ) t. Replacig the summa i (7) with P(φ( x) = v), we obtai Claim. For each v V a 3 i t +, defie θ i (v) by g a(i) (v) = d(v) i + 2 (d i + 2) θ i(v). (0) Note that sice g l(v) = P(φ(x l) = v) = for all l t +, we have (d(v) i + 2)θ i (v) = d i + 2 a this verifies (2). The goal of the ext claim is to verify (3). Claim 2. For 3 i t +, t ) t θi (v) k ) t. k t Proof of Claim 2. We oly prove the upper bou, sice the lower bou is similar. For coveiece, for 3 l i, a v = u = u b(2), defie f(l) = u i N + (u b(i) ) u 2 N + (u b(2) ) u 3 N + (u b(3) ) l u l N + (u b(l) ) j=3 d(u b(j) ) t. The g i (v) f(i) by Claim. Sice N + (u b(i) ) d(u b(i) ) a d(u b(i) ) k, d(u b(i) ) t = N +(u b(i) ) d(u b(i) ) t d(u b(i)) d(u b(i) ) t k k t. This shows f(i) k k t f(i ). Cotiuig i this way we obtai ) i 2f(2) g i (v) f(i) k t = ) i 2 k t = d(v) k t d(v) k t u 2 N(v) ) i 2 ) t 2. the electroic joural of combiatorics 22 (205), #P00 6

7 For ay j such that 3 j i t +, d(v)/() k/(k j + 2) k/(k t) sice d(v) k t + j. Therefore d(v) (d j + 2) d(v) (d j + 2) k k t. Isertig this i the precedig upper bou for g i (v), we fi for ay j with 3 j i t +, g i (v) (d j + 2) k ) t 2 k t ) t. k t (d j + 2) k t This also holds for i = 2 as g 2 (v) = d(v) d (d j + 2) d(v) Replacig i with a(i) 2 a selectig j = i, g a(i) (v) d(v) i + 2 ) t. (d i + 2) k t ). (d j + 2) k t By the defiitio (0), this gives θ i (v) (k/(k t)) t. The lower bou is similar: we use the lower bou o g i (v) from Claim, together with the observatio N + (u b(i) ) d(u b(i) ) t a d(u b(i) ) k to obtai The we use d(v) d g i (v) d(v) t (d j + 2) d j + 2 d k ) t 2. (d j + 2) k t k for 3 j t +, a as above this gives the required lower bou o θ i (v). Now we fiish the proof of Theorem 3: (5) becomes N T (G) t+ i=3(d(v) i + 2) ga(i)(v) = t+ (d(v) i + 2) d(v) i+2 where θ i (v) satisfies (2) a (3), by Claims a 2. This proves (). i=3 (d i+2) θ i(v) 3 A Approximate Jese s Iequality A approximate form of Jese s Iequality is used to prove Theorem from Theorem 3: the electroic joural of combiatorics 22 (205), #P00 7

8 Lemma 4. Let δ, x i, y i [0, ) for all i [], a suppose i= x i = i= y i = γ a x i /y i δ for all i []. The i= y i γ xi ( δ)e δ γ. Proof. The lemma is clear if δ so we assume δ [0, ). This implies x i, y i (0, ). Let f(y) = y log y a x i y i = c i. Sice f is covex o (0, ), Jese s Iequality gives: ( ) f(y i ) f y i = γ log γ. i= i= Furthermore, y i log x i = f(y i ) + y i log c i. Therefore log i= y i γ xi = γ y i log x i i= = γ = ( γ (f(y i ) + y i log c i ) i= i= log γ + γ = log γ + γ ) f(y i ) + γ y i log c i i= i= y i log c i i= y i (c i ) + γ y i (log c i c i + ). Sice c i y i = x i = γ, y i (c i ) = 0. The fuctio h(x) = log x x o the iterval [ δ, + δ] has oe critical poit, a maximum at x =, a therefore its miimum i this iterval occurs at oe of its epoits. We claim that h( δ) < h( + δ). Ieed, this is equivalet to e 2δ < ( + δ)/( δ) = + j= 2δj. Sice e 2δ = + j= (2δ)j /j! a (2δ) j /j! 2δ j for all j with strict iequality for j > 2, this proves that h( δ) < h( + δ) for δ [0, ). Therefore log c i c i + log( δ) + δ for all i [], which gives log i= Expoetiatig gives the lemma. i= y i y xi log γ + log( δ) + δ. Remarks. As δ 0, ( δ)e δ = 2 δ2 + O(δ 3 ). If is eve, y i = γ for all i a x i = ( + δ)γ for i a x 2 i = ( δ)γ for i >, the as δ 0, 2 i= y i γ xi = ( δ 2 ) 2 γ = 2 δ2 + O(δ 3 ). the electroic joural of combiatorics 22 (205), #P00 8

9 This shows the bou i Lemma 4 is asymptotically tight as δ 0. Note that if δ = 0, the i= xy i/γ i γ directly from Jese s Iequality. 4 Proof of Theorem We prove Theorem by iuctio o. Let G be a -vertex graph with average degree d t. Theorem states N T (G) ( ε d )(d) t. If t d < 6t 2, the ε d 0 a so Theorem is true i this case. This also proves the theorem whe < 6t 2. Suppose d 6t 2. If k is the miimum degree of G, we cosider the two cases k < d/8 a k d/8 separately. Case. k < d/8. I this case, remove from G a vertex of degree k to get a graph H with vertices a average degree at least 2 d ( 2 d ) > d + d 8 d 4( ) > d + 3d 4 := d. By defiitio of d, ( )d (d ) = ( )(d + 3d 3d )(d = (d ) + d d2 9d2 + 3d > (d ) + 2 d2 d2 > (d ). From the seco to the third lie, we used 3d/4 > 9d 2 /6 2 sice d <, a 5d 2 /6 < d 2 /. Recall ε d = (4t) 5 /d 2. Note also d > d t, so by iuctio, a sice ε d ε d for d d, The proof is complete i this case. N T (H) ( ε d ) ( )(d ) t ( ε d ) ( )d (d ) (d 2) t 2 > ( ε d ) (d ) (d 2) t 2 = ( ε d ) (d) t. Case 2. k d/8. We apply Lemma 4 a Theorem 3. By Theorem 3, shiftig the rage of summatio from 3 i t + to j [t ], N T (G) t (d(v) j) d(v) j (d j) θj+2(v). () j= By (3), for all j [t ], t ) t ) t. k θ j+2 (v) k t the electroic joural of combiatorics 22 (205), #P00 9

10 Sice ( x) t xt for 0 x a ( + x) t + 2xt for 0 x /t, t2 k θ j+2 (v) + 2t2 k t. Sice 2t 2 /(k t) 4t 2 /k is easily true for k d/8 2t 2, lettig δ = 4t 2 /k we obtai θ j+2 (v) δ for all j [t ]. Now we use Lemma 4. Fixig j [t ], let x v,j = d(v) j a let y v,j = (d(v) j)θ j+2 (v) for v V. The Furthermore, from (2) i Theorem 3, x v,j = y v,j θ j+2 (v) δ. y v,j = (d(v) j)θ j+2 (v) = d j = (d(v) j) = x v,j. Also x v,j d(v) j k t 0 a y v,j = θ j+2 (v)x v,j 0. Therefore all the coitios of Lemma 4 are satisfied, a so for each j [t ], Lemma 4 with γ = d j gives x y v,j (d j) v,j Applyig this for each j [t ] to () gives ( δ)e δ (d j). t N T (G) ( δ)e δ (d j) = (d) t ( δ) t e (t )δ. j= Fially, usig ( δ)e δ δ 2 a ( δ 2 ) t tδ 2, ( δ) t e (t )δ ( δ 2 ) (t ) tδ 2 = 6t5 k 2 ε d. From the third lie to the fourth, we used k d/8. This completes the proof. the electroic joural of combiatorics 22 (205), #P00 0

11 5 Cocludig remarks Theorem ca be improved if structural iformatio o G is give. For istace, if G is a -vertex triagle-free graph of average degree d, a T is the t-edge path, the oe ca adapt the proof of Theorem to obtai t N T (G) ( O( t5 )) (d i ). d 2 2 A disjoit uio of complete bipartite graphs K d,d shows this is tight up to the factor O( t5 ). It is plausible as i Cojecture that this costructio has the fewest copies d 2 of T amogst all triagle-free graphs of average degree d, whe d is large eough. Fix a tree T with t edges a degrees, d, d 2,..., d t i a breadth-first labellig startig with a leaf. A local isomorphism of T i a graph G is a eighbourhood preservig embeddig of T i G i.e. a map φ : V (T ) V (G) such that if φ(x) = v for some x V (T ), the φ(y) φ(z) wheever y, z N T (x) are distict. Usig the method of Theorem 3, oe ca show that the umber of local isomorphisms of T i G is at least i= t (d(v) ) d(v) d i= If G has miimum degree say at least 2t, a average degree d, it is possible to show that the above expressio is at least t (d ) di. i= If T is a path, the we recover the mai result of [] statig that the umber of obacktrackig walks of t edges i G is at least (d ) t. Furthermore, the bou is tight sice equality is achieved for ay d-regular -vertex graph. Fially, if G has girth at least diam(t ), the every local isomorphism of T i G is a isomorphism, so we obtai the lower bou t N T (G) (d ) di. i= d i. Agai, this is tight for ay d-regular graph of sufficietly large girth. Ackowledgmet. We thak Eoi Log for poitig out a error i a earlier versio of this paper, a the aoymous referees for helpful commets. Refereces [] N. Alo, S. Hoory, a N. Liial, The Moore Bou for Irregular Graphs, Graphs Combi. 8 (2002), o., the electroic joural of combiatorics 22 (205), #P00

12 [2] G. Blakley a P. Roy, Hölder type iequality for symmetrical matrices with oegative etries, Proc. Amer. Math. Soc. (965) 6, [3] D. Dellamoica Jr., P. Haxell, T. Luczak, D. Mubayi, B. Nagle, Y. Perso, V. Rödl, M. Schacht, Tree-miimal graphs are almost regular, Joural of Combiatorics 3 (202), o., [4] P. Erdős a M. Simoovits, Compactess results i extremal graph theory, Combiatorica 2 (982) o. 3, [5] A. F. Sidoreko, Iequalities for fuctioals geerated by bipartite graphs, Diskretaya Matematika 3 (99), (i Russia), Discrete Math. Appl. 2 (992), (i Eglish). [6] H. Täubig. J. Weihma, Matrix power iequalities a the umber of walks i graphs, Discrete Applied Mathematics 76, the electroic joural of combiatorics 22 (205), #P00 2

Counting trees in graphs

Counting trees in graphs Coutig trees i graphs Dhruv Mubayi Jacques Verstraëte April 8, 206 Abstract Erdős ad Simoovits proved that the umber of paths of legth t i a -vertex graph of average degree d is at least ( δ)d(d ) (d t