Counting trees in graphs

Transcription

1 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 + ), where δ = (log d) /2+o() as d. I this paper, we stregthe ad 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 ad miimum degree greater tha t, the umber of labelled copies of T i G is at least ( ε)d(d ) (d t + ) where ε = O(d 2 ) as d. This boud is tight except for the term ε, as show by a disjoit uio of cliques. Our proof is obtaied by first showig a lower boud that is a covex fuctio of the degree sequece of G, ad this aswers a questio of Dellamoica et. al. Itroductio If H ad 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 ad let d G (v) = N G (v). Let (a) b = a(a )(a 2)... (a b + ) whe a b ad a is real ad b is a iteger, ad let (a) 0 =. Let T be a t-edge tree ad G = (V, E) be a graph with miimum degree at least t. I this paper, we obtai lower bouds o N T (G). This is a basic questio i combiatorics, for example, the simple lower boud (d(v)) t i the case whe T is a star is the mai iequality eeded for a variety of fudametal problems i extremal graph theory. Coutig paths ad walks i graphs has umerous applicatios, such as fidig bouds o the spectral radius of the graph ad the eergy of a graph (see Täubig ad Weihma [6] ad the refereces therei), as well as i stadard graph theoretic reductios of combiatorial problems such as coutig words over alphabets with forbidde patters. Blakley ad Roy [2] used elemetary liear algebraic methods to show that if A is ay symmetric poitwise o-egative matrix ad v is ay poitwise o-egative uit vector, the for ay t N, Departmet of Mathematics, Statistics, ad 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

2 A t v, v Av, v t. If t is eve, this is a simple cosequece of the fact that A is diagoalizable ad 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 d 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 ad the tesor power trick [5], as well as geeralizatios to various other iequalities ivolvig walks (see [6] ad the refereces therei). Alo, Hoory ad Liial [] showed that the umber of o-backtrackig walks i a -vertex graph G of average degree d ad miimum degree at least two is at least d(d ) t, which is agai tight for d-regular graphs. A key fact i their proof is that if oe cosiders a radom walk of legth t o the graph, startig with a uiformly radomly chose edge, the the distributio of every edge of the walk is also uiform. The method of Alo, Hoory ad 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 d(v)(t )d(v)/d. By Jese s Iequality, this is at least d t whe G has o isolated vertices. I Problem 4 of [3], the questio of fidig similar lower bouds 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 (d 2)2d(v)/d whe G has miimum degree at least three. If P is a t-edge path, ad G is a d-regular graph, the clearly N P (G) (d) t with equality for d t if ad oly if every compoet of G is a clique. Erdős ad 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], exted the result i [4] to arbitrary trees ad stregthe their lower boud to (d O(/d)) t as d : Theorem. If T is ay t-edge tree ad G is a -vertex graph of average degree d ad miimum degree greater tha t, ad ε = (4t) 5 /d 2, the N T (G) ( ε)(d) t. This result is best possible up to the factor ε 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 ad average degree d for which N T (G) < (d) t, we propose the followig cojecture: Cojecture. For every t >, there exists d 0 = d 0 (t) such that for ay tree T with t edges ad ay -vertex graph G with average degree d d 0, N T (G) (d) t. 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. 2

3 This paper is orgaized as follows. I the ext sectio, we determie a lower boud 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 exteds the method of [] by carefully cotrollig the distributio of the edges i a radom embeddig of a tree i a graph of average degree d. The fuctio of degrees is almost covex, ad 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 ad coutig trees As a steppig stoe to Theorem, we are goig to prove the followig theorem, which gives a lower boud for N T (G) i terms of the degree sequece of G, ad addresses Problem 4 i [3]: Theorem 2. Let T be a t-edge tree, ad let G = (V, E) be a -vertex graph with average degree d ad miimum degree k > t >. The N T (G) d t+ (d(v) i + 2) d(v) i+2 i=3 (d i+2) θ i(v) () where for 3 i t +, ad for 3 i t +, (d(v) i + 2)θ i (v) = d i + 2 (2) ( k t ) t θi (v) k ( 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) = ad a(3) = 2. Let v v 2 be a radomly ad uiformly chose (orieted) edge of G ad defie φ(x ) = v ad 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 ad therefore N + (v a(i) ) N(v a(i) ) {v, v 2,..., v i }\{v a(i) } d(v a(i) ) i + 2 > 0. 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 3

4 evet that φ(x i ) = v i for all i [t + ]. The By the iequality of arithmetic ad geometric meas: P(φ( x) = v) = t+ d 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) = (d) P(φ( x)= v) t+ N + (v a(i) ) P(φ( x)= v) v Ω v Ω i=3 = d 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) d t+ N + (v a(i) ) P(φ( x)= v) v Ω i=3 t+ = d i=3 t+ d i=3 t+ = d i=3 t+ = d i=3 v Ω iv N + (v a(i) ) P(φ( x)= v) I the ext two claims, we fix v V. For i [t + ], defie v Ω iv (d G (v) i + 2) P(φ( x)= v) v Ω P(φ( x)= v) (d G (v) i + 2) iv (d G (v) i + 2) P(φ(xa(i))=v). (5) 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 ad lower bouds o g i (v) relative to d G (v)/d. We write d(v) istead of d G (v) i what follows. To boud 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 ad 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 4

5 y, y 2,..., y i ad 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 ad N + (u j ) = N G (u j )\{u, u 2,..., u j }. Claim. Let 3 i t + ad let v = u b(2) = u. The d u 2 N + (u b(2) ) u i N + (u b(i) ) j=3 d(u b(j) ) g i(v) d 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 u 2 N + (u b(2) ) u 3 N + (u b(3) ) u i N + (u b(i) ). (7) 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 ad 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 probabilty Sice d(v a(j) ) t N + (v a(j) ) d(v a(j) ), d j=3 P(φ( x) = v) = d j=3 P(φ( x) = v) d(v a(j) ) d N + (v a(j) ). (8) j=3 d(v a(j) ) t. (9) We ow prove {v a(j) : 3 j i} ad {u b(j) : 3 j i} are equal as multisets. For example, if T i is a path, the these multisets are sets, ad {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 umber of times u l is a paret of some vertex v j i the orderig v,..., v i, ad this is agai d T (y l ). We coclude {v a(j) : 3 j i} ad {u b(j) : 3 j i} are equal as 5

6 multisets. Therefore by (9), d j=3 P(φ( x) = v) d(u b(j) ) d j=3 d(u b(j) ) t. Replacig the summad i (7) with P(φ( x) = v), we obtai Claim. For each v V ad 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 ad this verifies (2). The goal of the ext claim is to verify (3). Claim 2. For 3 i t +, ( k t ) t θi (v) k ( k ) t. k t Proof of Claim 2. We oly prove the upper boud, sice the lower boud is similar. For coveiece, for 3 l i, ad v = u = u b(2), defie f(l) = d 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) ) ad d(u b(i) ) k, This shows f(i) u i N + (u b(i) ) 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 tf(i ). Cotiuig i this way we obtai g i (v) f(i) ( k ) i 2f(2) k t = ( k ) i 2 d k t = d(v) ( k d k t d(v) ( k d k t 6 u 2 N(v) ) i 2 ) t 2. k k t.

7 For ay j such that 3 j i t +, d(v)/(d(v) j + 2) k/(k j + 2) k/(k t) sice d(v) k t + j. Therefore d(v) d d(v) j + 2 (d j + 2) d(v) d(v) j + 2 d(v) j + 2 (d j + 2) k k t. Isertig this i the precedig upper boud for g i (v), we fid for ay j with 3 j i t +, This also holds for i = 2 as g i (v) d(v) j + 2 (d j + 2) k ( k ) t 2 k t d(v) j + 2 ( k ) t. k t (d j + 2) k t g 2 (v) = d(v) d d(v) j + 2 (d j + 2) d(v) d(v) j + 2 d(v) j + 2 ( k ). (d j + 2) k t Replacig i with a(i) 2 ad selectig j = i, g a(i) (v) d(v) i + 2 ( k ) t. (d i + 2) k t By the defiitio (0), this gives θ i (v) (k/(k t)) t. The lower boud is similar: we use the lower boud o g i (v) from Claim, together with the observatio N + (u b(i) ) d(u b(i) ) t ad d(u b(i) ) k to obtai g i (v) d(v) ( k t ) t 2. d k The we use d(v) d d(v) j + 2 (d j + 2) d j + 2 d(v) j + 2 d (d j + 2) k t k for 3 j t +, ad as above this gives the required lower boud o θ i (v). Now we fiish the proof of Theorem 2: (5) becomes t+ N T (G) d i + 2) i=3(d(v) ga(i)(v) = d where θ i (v) satisfies (2) ad (3), by Claims ad 2. This proves (). t+ (d(v) i + 2) d(v) i+2 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 2: Lemma 3. Let δ, x i, y i [0, ) for all i [], ad suppose i= x i = i= y i = γ ad x i /y i δ for all i []. The i= y i γ xi ( δ)e δ γ. 7

8 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 ad x i y i = c i. Sice f is covex o (0, ), Jese s Iequality gives: i= ( f(y i ) f 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 ( ) y i = γ log γ. i= (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 =, ad therefore its miimum i this iterval occurs at oe of its edpoits. We claim that h( δ) < h( + δ). Ideed, this is equivalet to e 2δ < ( + δ)/( δ) = + j= 2δj. Sice e 2δ = + j= (2δ)j /j! ad (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 Expoetiatig gives the lemma. log i= 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 ad x i = (+δ)γ for i 2 ad x i = ( δ)γ for i > 2, the as δ 0, i= y i γ xi = ( δ 2 ) 2 γ = 2 δ2 + O(δ 3 ). This shows the boud i Lemma 3 is asymptotically tight as δ 0. Note that if δ = 0, the γ directly from Jese s Iequality. i= xy i/γ i 8

9 4 Proof of Theorem We prove Theorem by iductio o. Let G be a -vertex graph with average degree d ad miimum degree k. Theorem states N T (G) ( ε d )(d) t. If d < 6t 2, the ε d 0 ad so Theorem is true i this case. This also proves the theorem whe < 6t 2. Suppose d 6t 2. We cosider the two cases k < d/8 ad 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 ad average degree at least The 2 d ( 2 d ) > d + d 8 d 4( ) > d + 3d := c. 4 ( )c(c ) = ( )(d + 3d 3d 4 )(d + 4 ) = d(d ) + 4 d + 2 d2 9d2 + 3d d2 6 > d(d ) + 2 d2 d2 > d(d ). From the secod to the third lie, we used 3d/4 > 9d 2 /6 2 sice d <, ad 5d 2 /6 < d 2 /. By iductio, ad sice ε c ε d for c d, The proof is complete i this case. N T (H) ( ε c ) ( )(c) t ( ε c ) ( )c(c ) (d 2) t 2 > ( ε c ) d(d ) (d 2) t 2 ( ε d )(d) t. Case 2. k d/8. We apply Lemma 3 ad Theorem 2. By Theorem 2, shiftig the rage of summatio from 3 i t + to j [t ], By (3), for all j [t ], N T (G) d t (d(v) j) d(v) j (d j) θ j+2(v). () j= ( k t ) t ( k ) t. k θ j+2 (v) k t Sice ( x) t xt for 0 x ad ( + x) t + 2xt for 0 x /t, t2 k θ j+2 (v) + 2t2 k t. 9

10 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 3. Fixig j [t ], let x v,j = d(v) j ad let y v,j = (d(v) j)θ j+2 (v) for v V. The Furthermore, from (2) i Theorem 2, 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 ad y v,j = θ j+2 (v)x v,j 0. Therefore all the coditios of Lemma 3 are satisfied, ad so for each j [t ], Lemma 3 with γ = d j gives x y v,j (d j) v,j ( δ)e δ (d j). Applyig this for each j [t ] to () gives t N T (G) d ( δ)e δ (d j) = (d) t ( δ) t e (t )δ. j= Fially, usig ( δ)e δ δ 2 ad ( 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. 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, ad T is the t-edge path, the oe ca adapt the proof of Theorem to obtai t N T (G) ( O( t5 ))d (d i d 2 2 ). i= 0

11 A disjoit uio of complete bipartite graphs K d,d shows this is tight up to the factor O( t5 d 2 ). It is plausible as i Cojecture that this costructio has the fewest copies of T amogst all triagle-free graphs of average degree d, whe d is large eough. Fix a tree T with t edges ad 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 2, oe ca show that the umber of local isomorphisms of T i G is at least d t (d(v) ) d(v) d i= If G has miimum degree say at least 2t, ad average degree d, it is possible to show that the above expressio is at least t d (d ) di. i= If T is a path, the we recover the mai result of [] statig that the umber of o-backtrackig walks of t edges i G is at least d(d ) t. Furthermore, the boud 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 boud d i. t N T (G) d (d ) di. Agai, this is tight for ay d-regular graph of sufficietly large girth. i= Ackowledgmet. We thak Eoi Log for poitig out a error i a earlier versio of this paper. Refereces [] N. Alo, S. Hoory, ad N. Liial, The Moore Boud for Irregular Graphs, Graphs Combi. 8 (2002), o., [2] G. Blakley ad P. Roy, Hölder type iequality for symmetrical matrices with o-egative 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 ad 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).

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