The poset on connected graphs is Sperner

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1 The poset o coected graphs is Sperer Stephe GZ Smith Istvá Tomo February 9, 017 Abstract Let C be the set of all coected graphs o vertex set [] The C is edowed with the followig atural partial orderig: for G, H C, let G H if G is a subgraph of H The poset (C, is graded, each level cotaiig the coected graphs with the same umber of edges We prove that (C, has the Sperer property, amely that the largest atichai of (C, is equal to its largest sized level This aswers a questio of Katoa Keywords: Sperer's Theorem, Coected Graphs, Posets 1 Itroductio Let (P, be a partially ordered set (poset We oly cosider partially ordered sets with itely may elemets A chai i P is a set C P of pairwise comparable elemets A atichai A P is a set of pairwise icomparable elemets The poset (P, is graded if there exists a partitio of P ito subsets A 0,, A m such that A 0 is the set of miimal elemets of P, ad wheever x A i ad y A j with x < y ad there is o z P with x < z < y, the we have j = i + 1 If such a partitio exists, it is uique ad the sets A 0,, A m are the levels of P A graded poset (P, is Sperer if the largest atichai i P is the largest sized level Let m be a positive iteger, [m] = {1,, m} The Boolea lattice [m] is the power set of [m] ordered by iclusio, ad [m] ( = {A [m] : A = } By the well ow theorem of Sperer [9], the poset ( [m], is Sperer, the largest atichais beig equal to [m] ( m/ ad [m] ( m/ The questio whether certai posets are Sperer is widely studied For a short list of such results, see [1] I this paper, we ivestigate the Sperer property of the followig poset Let be a positive iteger ad let C deote the set of all coected graphs o vertex set [] (I other words, C is the family of labeled coected graphs o vertices The family C is edowed with the followig atural partial orderig: for G, H C, let G H if G is a subgraph of H, or more formally, if E(G E(H Whe there is o ris of cofusio, we shall simply write C whe referrig to the poset (C, Observe that C is graded, the levels of C beig the families C ( = {G C : E(G = } for = 1,, m The followig questio origiates from Katoa [5] Questio 1 Is (C, Sperer? Departmet of Mathematical Scieces, Uiversity of Memphis, Memphis, TN 3815, USA sgsmith1@memphisedu Departmet of Pure Mathematics ad Mathematical Statistics, Uiversity of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK itomo@camacu 1

2 We prove that the aswer is yes More precisely, settig m = ( ad M = m/, the mai result of this paper is the followig theorem Theorem If is sucietly large, the uique largest atichai i C is C (M Let us mae a remar about how this result compares to Sperer's theorem [9] Let G be the set of all graphs o vertex set [] ad exted the orderig to G i the obvious way Also, for = 0,, m, let G ( be the set of graphs i G with edges Observe that (G, < is isomorphic to ( [m],, hece (G, < is Sperer Note that C is a very dese subset of G As we shall see i Sectio 3, the size of C is at least m (1 o( This correspods to the well ow statemet that a graph chose uiformly at radom amog all graphs with vertices (that is a elemet of G(, 1/ i the Erd s-réyi radom graph model is discoected with a probability that is expoetially small A problem similar to Questio 1 has bee cosidered i a paper of Jacobso, Kézdy ad Seif [4] Let G be a coected graph ad let (C(G, < be the poset, whose elemets are the coected, vertex-iduced subgraphs of G, ad H < H if H is a iduced subgraph of H I [4], it was proved that this poset eed ot be Sperer, eve if G is a tree This paper is orgaized as follows I Sectio, we discuss our otatio ad prove a few techical results I Sectio 3, we shall prove various bouds o the umber of coected graphs with certai properties These bouds provide us with some of the igrediets eeded for the proof of Theorem i Sectio 4 I Sectio 5, we propose some ope problems Prelimiaries Let us say a few words about our otatio, which is mostly covetioal If G is a graph, V (G is the vertex set of G, E(G is the set of its edges, ad e(g = E(G If U V (G, G[U] deotes the subgraph of G iduced o the vertex set U If F E(G, the G F is the graph o vertex set V (G ad edge set E(G \ F If e E(G, we simply write G e istead of G {e} For the sae of readability, we use the otatio exp (x = x, whe ecessary Furthermore, log deotes base logarithm Our paper cotais a lot of techical computatios that are made more coveiet by the followig extesio of the biomial coeciet We dee the biomial coeciet ( x for ay N, x R such that ( { x(x 1(x +1 x =! if x, 0 otherwise We collect some of the simple properties of ( x i the followig lemma Lemma 3 Let N, x R (i If x, we have ( ( x 1 / x = x +1 (ii Let δ be a o-egative iteger ad suppose that x δ The ( x+δ (iii ( ( x x+1 ( ad x ( x+1 +1 δ ( x

3 (i ad (iii easily follows from the deitio Now let us prove (ii If we prove the case δ = 1, that is ( ( x+1 x for x 1, the result follows by iductio o δ But i this case, we have ( ( x+1 / x = (x + 1/(x + 1 We remar that by cotiuity, for ay xed positive iteger ad a real umber r 1, there is a uique x R such that r = ( x Throughout this paper, we shall also use the followig simple iequalities Lemma 4 Let a 1,, a s be positive itegers ad let a a s = We have s ( ai ( s + 1, (i ad 1 i<j s ( s 1 a i a j ( s + 1(s 1 + (ii Also, if a i for i [s], where / < s + 1, the s ( ai ( s + + (, (iii ad 1 i<j s a i a j ( (iv The fuctio f(x = x is covex, so s a i attais its maximum uder the coditios s a i = ad a i Z + whe a 1 = = a s 1 = 1 ad a s = s + 1 Note that the left had side of (i is s a i / /, ad the left had side of (ii is ( s a i /, while the right had sides of these iequalities are the respective values whe a 1 = = a s 1 = 1 ad a s = s + 1 For the iequalities (iii ad (iv, otice that with the additioal coditio that a i, s a i attais its maximum whe a 1 = = a s = 1, a s 1 = s +, a s = The right had side of (iii is exactly s ( ai with these values iserted O the other had, we have a i a j a s (a a s 1 = a s ( a s = (, which proves (iv 1 i<j s 3

4 3 Coectivity of graphs Figure 1: A graph ad a illustratio of its bloctree I this sectio, we ivestigate the followig problems How may edges ca a graph G have, whose removal destroys the coectivity, or -edge-coectivity of G? Also, what is the umber of -edge-coected graphs G o vertex set [] i which there are exactly r edges, whose removal destroys the -edge-coectivity of G? Let us start this sectio with the followig well ow result about the umber of discoected graphs For completeess, we shall provide a short proof A stroger form of this result ca be foud i [], p 138 as well ( ( 1 Lemma 5 The umber of discoected graphs o vertex set [] is less tha exp + o( A graph G is discoected if there is a partitio of [] ito two oempty sets A ad B such that there are o edges betwee A ad ( B The umber of discoected graphs, where A = 1 ( 1 ad B = 1 is at most exp, as we have choices for the partitio {A, B}, ad ( ( 1 exp umber of dieret choices for the edges i B ( The umber of discoected graphs where A, B is at most exp + ( + 1 = ( ( 1 exp + 3, as there are at most exp ( umber of choices for the partitio (A, B, ad the ( ( A ( umber of ways to choose the edges iside A ad B is at most exp + B ( ( exp +1 ( ( 1 Hece, the total umber of discoected graphs is at most exp + o( We dee the bloc tree of a coected graph G as follows A edge e E(G is a bridge, if G e is discoected Let B be the set of bridges i G ad let A 1,, A t be the vertex sets of the compoets of G B The the bloc tree of G is Bt(G = (B, {A 1,, A t } The followig lemma lists the mai properties of the bloc tree, which may be easily veried by the reader 4

5 Lemma 6 Let G be a coected graph with bloc tree (B, {A 1,, A t } The B = t 1 ad G[A i ] is -edge-coected for i [t] If G is a -edge-coected graph, let R(G be the set of edges f E(G such that G f is ot -edge-coected Lemma 7 gives a upper boud o the size of R(G Lemma 7 Let G be a -edge-coected graph ad let H = G R(G compoets of G R(G by q The R(G q Deote the umber of To mae our proof more coveiet, we shall wor with multi-graphs A multi-graph is a graph where we allow multiple edges betwee a pair of vertices, but o loops We exted the deitio of a cycle as follows: a cycle is either vertices coected by edges or a simple graph that is a cycle A chord i a cycle C is a edge ot i E(C coectig two vertices of C For example, if the vertices x ad y are coected by 3 edges, ay two edges form a cycle ad the third edge is a chord of this cycle Let the compoets of H be H 1,, H q Every edge i R(G coects two dieret compoets i H Dee the multi-graph K o vertex set [q] as follows: if H i ad H j are coected by l edges i G, the i ad j are coected by l edges i K Note that the graph K caot cotai a cycle with a chord Otherwise, suppose that there is a cycle with vertices i 1,, i s ad a chord i a i b Let e R(G be a edge coectig H ia ad H ib i G The H ia ad H ib are still coected by at least disjoit paths i G e, hece e caot be a elemet of R(G Our lemma follows from the followig result about multi-graphs without cycles with a chord Claim 8 If L is a multi-graph o q vertices without a cycle with a chord, the e(l q We proceed by iductio o q If q = 1, E(L is empty, so we are doe Suppose that q > 1 If L has a vertex v of degree at most, the let L = L v The L has q 1 vertices, at least e(l edges, ad does ot cotai a chorded cycle Hece, by iductio, e(l q 4, which gives e(l q Now suppose that every vertex of L has degree at least 3 Let v 1,, v s be the cosecutive vertices of a logest path i L Every eighbor of v 1 is cotaied i the set {v,, v s }, otherwise we ca d a loger path i L Hece, there exist i, j satisfyig i j s such that the multi-set E(L cotais three dieret edges, v 1 v, v 1 v i ad v 1 v j But the v 1,, v j forms a cycle, ad v 1 v i is a chord of this cycle As K does ot cotai a cycle with a chord ad has R(G edges, we get R(G q This completes the proof of Lemma 7 We remar that if R(G is o-empty, it has at least elemets This is true because if e R(G, the G e cotais a bridge f But the f R(G as well Let I r be the set of -edge-coected graphs G such that R(G = r, ad let I r ( = I r C ( I the ext lemma, we give a upper boud o the size of I r ( Recall that M = ( / 5

6 Lemma 9 Let ɛ be a positive real umber There exists 1 (ɛ such that if > 1 (ɛ, the followig holds For ay positive itegers r ad satisfyig r ad M M +, we have I r ( (( r/ + ɛr Let q = r/ + 1 If G is a -edge-coected graph with R(G = r, the G R(G has at least q compoets by Lemma 7 We ow cout the umber of graphs G where G R(G has exactly s compoets Note that s r, otherwise the edges of R(G could ot coect all the compoets of G R(G The umber of graphs G, for which R(G = s, ad where the compoets i G R(G have sizes a 1,, a s with e 1,, e s edges iside them, respectively, is at most ( ( r a 1,, a s s (( ai e i (1 Here, ( ( r is a upper boud o the umber of ways to pic the edges of R(G, a 1,,a s is the umber of ways to partitio [] ito parts of size a 1,, a s, ad ( ( a i e i is the umber of ways to choose the e i edges i a compoet of size a i We shall prove that (1 is at most ( ( s+1 exp (3ɛr/6 Let us boud the terms i (1 First, ( r exp (r log < exp (ɛr/6, if is sucietly large give ɛ Also, ( a 1,,a s s = exp ( log s Ufortuately, if r is small, we caot boud this term by exp (cɛr, where c is some xed costat We shall overcome this obstacle later i the proof Fially, s (( ai ( s ( ai r where the last iequality holds by (i i Lemma 4 Here, see (iii i Lemma 3 Hece, we have e i < r (( s+1 s (( ai e i ( ( r + s+1 (( s+1 (( s+1 r, + ɛr/6 provided > 6/ɛ First, suppose that r is such that log r < ɛr/6 I this case, we have ( a 1,,a s exp (ɛr/6 Hece, (1 is at most ( ( s+1 +ɛr/6 exp (ɛr/6 Now cosider the case whe log r > ɛr/6 The r < R(ɛ, where R(ɛ is a costat oly depedig o ɛ I this case, we shall boud the product ( a 1,, a s s (( ai e i,, ( 6

7 Without loss of geerality, suppose that a 1 a s ad observe that ( a 1,,a s < a ++a s Thus, if a 1 4r, the ( a 1,,a s < 4r < exp (ɛr/6, if is sucietly large give ɛ Now suppose that a 1 < 4r Applyig (iii i Lemma 4, we get s ( ai ( 4r s + ( 4r 8r + ( 4r Suppose > 0R(ɛ, the the iequality ( ( 4r s + 1 8r + r holds as well Hece, s (( ai ( s ( ai < r e i (( s+1 r < r (( s+1 r + r where the last iequality holds by (iii i Lemma 3 Also, usig (ii i Lemma 3, (( s+1 (( r + r s+1 exp ( r Thus, we ca boud ( from above by ( ( s+1, ad so (1 is at most i this case as well (( s+1 + ɛr/6 exp (ɛr/6 Now let us boud the umber of all -edge-coected graphs with edges, for which R(G = r ad G R(G has s compoets The umber of such graphs is at most a 1 ++a s= e 1 ++e s= r ( ( r a 1,, a s s (( ai e i, (3 The rst sum has exactly ( s 1 terms sice ai 1 for every i [s], while the secod sum has terms Therefore, (3 is at most ( r+s s 1 Here, ( s 1 ( ( (( r + s s+1 + ɛr/6 exp s 1 s 1 (ɛr/6 exp (r log ad ( r+s < exp (r log Thus, (3 is at most s 1 (( s+1 + ɛr/6 exp (3ɛr/6, provided is sucietly large give ɛ Fially, the umber of -edge-coected graphs with R(G = r ad edges is at most 7

8 r i=q (( i+1 + ɛr/6 Applyig (ii i Lemma 3, we get exp (3ɛr/6 < I r ( (( q+1 (( q+1 + ɛr + ɛr/6 exp (4ɛr/6 I the proof of Theorem, we shall also use the followig techical lemma Agai, recall that M = ( / Lemma 10 Let > 150 Let G be a coected graph o vertex set [] such that e(g M ad Bt(G = (B, {A 1,, A t } Suppose that A i for i [t] The 1 i<j t A i A j (t 1 t R(G[A i ] (4 By Lemma 7, we have R(G[A i ] < A i Hece, t R(G[A i] < First, suppose that max{ A 1,, A t } 6 By (iv i Lemma 4, we have A i A j 6( i<j t Hece, usig the trivial boud t 1 <, we have that (4 holds Now suppose that A 1 5 I this case, we have t 6 Let H = G[A 1 ] Every edge of G ot cotaied i H is either i B or it is a edge of G[[] \ A 1 ] Hece, the umber of edges ot cotaied i H is at most 0, so e(h M 0 Let H 1,, H q be the vertex sets of the compoets of H R(H The, by Lemma 7, the umber of edges of H is at most q + q ( ( V (Hi q + 1 < +, where the iequality holds by (i i Lemma 4 Comparig the lower ad upper bouds o e(h we get the iequality ( q + 1 M 0 < + If q > /3, the right had side of the iequality is at most /9 + 3, while the left had side is larger tha /4 This is a cotradictio, otig that /9 + 3 < /4 for > 150 Hece, we have q < /3, implyig R(H < /3 This gives t R(G[A i ] R(H + ( A + + A t < / Sice A 1, we have 1 i<j t A i A j ( by (iv i Lemma 4, so (4 holds 8

9 4 Matchigs betwee levels I this sectio, we prove Theorem Let 1, l m We say that there is a complete matchig from C ( to C (l, if there is a ijectio f : C ( C (l such that G ad f(g are comparable for all G C ( The ext lemma states that to prove Theorem, it is eough to d a complete matchig from the smaller sized level to the larger sized level for ay two cosecutive levels Due to its simplicity, we shall oly setch the proof of this lemma Lemma 11 Suppose that there is a complete matchig from C ( to C (+1 for = 1,, M 1, ad there is a complete matchig from C (l+1 to C (l for l = M,, m 1 The the largest atichai i C is C (M Usig the complete matchigs, oe ca build a chai partitio of C ito C (M chais But the size of the maximal atichai i C is at most the umber of chais i ay chai partitio of C First, we show that if we are below the middle level C (M, or at least above the middle level, the it is easy to prove the existece of a complete matchig betwee cosecutive levels Let X C ( for some 1 m The lower shadow of X is ad the upper shadow of X is (X = {G C ( 1 : H X, G < H}, (X = {G C (+1 : H X, H < G} I our proofs, we shall apply the well ow theorem of Hall [3] Theorem 1 (Hall's theorem Let G = (A, B; E be a bipartite graph There is a complete matchig i G from A to B if ad oly if X Γ(X for all X A, where Γ(X deotes the set of vertices adjacet to some elemet of X First, let us deal with the levels below C (M Lemma 13 There is a complete matchig from C ( to C (+1 for = 1,, M 1 Let X C ( By Hall's theorem, it is eough to show that X (X Let B be the bipartite graph with vertex partitio (X, (X, ad the edges of B beig the comparable pairs If G X, the degree of G is m Also, if H (X, the degree of H is at most + 1 Let e be the umber of edges of B The, coutig e from X, ad the from (X, we have ad Hece, X (m = e, e (X ( + 1 X (X ( + 1/(m (X 9

10 Usig similar ideas, we ow show that if we are above the middle level by at least, the there is a matchig from C (+1 to C ( Lemma 14 There is a complete matchig from C (+1 to C ( for = M +,, m Let X C (+1 By Hall's theorem, it is eough to show that X (X Let B be the bipartite graph with vertex partitio (X, (X, ad the edges of B beig the comparable pairs If G (X, the the degree of G i B is at most m Now let G X If e E(G such that G e is ot a elemet of C, the e is a bridge of G However, by Lemma 6, the umber of bridges of G is at most 1 Hece, the degree of G is at least + Coutig the umber of edges of B two ways, we get X ( + E(B, ad Hece, (X (m E(B (X X + m 1 Provig that there is a matchig from C ( to C ( 1 for the values of that are slightly larger tha M is more dicult The remaider of this sectio is devoted to this problem Before showig the details, we briey outlie the strategy for showig that there exists a complete matchig from C ( to C ( 1, where M + 1 < M + Our goal is to show that for every X C (, we have (X X To accomplish this, we write X as Y Z, where Y is the set of -edge-coected graphs i X ad Z is the set of the o--edge-coected graphs i X We rst show that if the two sets, Y ad Z, do ot have roughly the same size, the the larger of the two has a lower shadow that is already larger tha X Now suppose that Y Z We show the existece of three fuctios c 1, c, c 3 : N R + satisfyig the followig properties: 1 (Y Y (1 + c 1 ( Y, (Z Z (1 + c ( Z, 3 if U is the set of -edge-coected graphs i (Y, the U Y (1 c 3 ( Y, 4 c 1 ( Y c ( Z c 3 ( Y, if Y Z Roughly, 1 ad state that the lower shadow of Y ad Z is slightly larger tha Y ad Z, respectively Now, we would lie to guaratee that (X = (Y (Z is also larger tha Y Z If this is ot the case, the we must have that (Y \ (Z is too small But ote that as (Z cotais oly o--edge-coected graphs, U is cotaied i (Y \ (Z Hece, U is a lower boud o the size of the set (Y \ (Z Thus, 3 tells us that (Y \ (Z caot be much smaller tha Y, ad property 4 guaratees (as we shall see later that we truly have Y Z (Y (Z 10

11 We remid the reader that G is the family of all graphs o vertex set [] For X C (, let (X = {H G ( 1 : G X, H < G} As (G, < is isomorphic to ( [m],, the Krusal-Katoa theorem [6, 7] tells us which subfamily of G ( of give size miimizes the lower shadow Istead of usig this, however, we use a weaer form of the Krusal- Katoa theorem, proved by Lovász [8] This aords us a computatioally more coveiet way to obtai a lower boud o the size of (X Lemma 15 (Lovász [8] Let X C ( be oempty ad let x be a real umber such that X = ( x The ( x (X 1 I particular, (X X x + 1 We remid the reader that we use the exteded deitio of biomial coeciets itroduced i Sectio, so both i the previous lemma ad i what comes, x eed ot to be a iteger i ( x Let D be the set of -edge-coected graphs i C ad let D ( = C ( D If X D (, the (X = (X Hece, we ca use Lemma 15 to get a lower boud for the size of (X I the ext lemma we show that if the size of X C ( is sucietly large, the we have (X X Lemma 16 Let ɛ > 0 There exists (ɛ such that if > (ɛ the followig holds Let M + 1 < M + ad let X = ( ( x, where x > 1 + ɛ We have (X > X By Lemma 15, (X ( x 1 Let D be the set of discoected graphs with 1 edges By Lemma 5, (( 1 D exp + o( Also, ( (( x 1 (X = (X \ D (X D exp 1 + o( Thus, we get ( ( (( x x 1 (X X exp 1 + o( = ( (( (( x x (( = exp 1 + o( > + ɛ exp + o( By (ii i Lemma 3, we have ( ( 1 +ɛ ( 1 ( 1 1 exp (ɛ Also, ( ( 1 1 = exp ( ( 1 + o( holds by Stirlig's formula Hece, we have (( ( 1 1 (X X exp + ɛ + o( exp ( + o( Thus, if is sucietly large give ɛ, (X > X 11

12 Now we show that if X is a set of -edge-coected graphs i C (, the the umber of -edge-coected graphs i the shadow of X caot be much less tha X Lemma 17 Let 0 < ɛ < 1/4 There exists 3 (ɛ such that if > 3 (ɛ, the followig holds Let M < < M + ad let X D ( Let X = ( x ad let r be a positive iteger satisfyig r < If x > ( (r+1/ + ɛr, the (X D ( 1 > 1 4r X Dee U = (X D ( 1 ad let B be the bipartite graph with vertex partitio (X, U, the edges beig the comparable pairs Every elemet of U has degree at most m +1 i B Also, the degree of a graph G i X is exactly R(G i B Let a be the umber of graphs i X with degree at most r 1 ad let a be the umber of graphs i D ( with R(G r + 1 The a < a ad by Lemma 9, we have a < ( ( (r+1/ +ɛr/, provided > 1 (ɛ/ Moreover, we have the followig bouds o the umber of edges of B: Hece, Here, m/ + 1, so ( r( X a e(b (m + 1 U U X a r m + 1 U X a m/ r r 4 m/ ( 1 If X > 8 3 a, we get U X 1 4r, usig that r 1 But ote that if is sucietly large give ɛ, the 8 3 < exp (ɛ/3, which meas that 8 3 a < (( (r+1/ + ɛr/ exp (ɛ/3 < (( (r+1/ where the secod iequality is a cosequece of (ii from Lemma 3 + ɛr < ( x, We remar that we do ot have to cosider the case whe r If X = ( x 1, the x, ad we ca always d r < satisfyig x > ( (r+1/ + ɛr This remar holds true for the upcomig lemmas as well I the ext lemma, we show that if X C ( is a set of o--edge-coected graphs, the the size of the shadow of X is slightly larger tha X Lemma 18 Let ɛ be a positive real umber such that ɛ < 1/ There exists 4 (ɛ such that if > 4 (ɛ, the followig holds Let be a positive iteger with M < < M + ad let X C ( \ D ( Let X = ( ( x ad let r be a positive iteger such that r < ad x > (r+1/ + ɛr The (X X > r/ 1

13 Dee the bipartite graph B betwee X ad U = (X as follows Let G X ad H (X be coected by a edge if H < G ad Bt(G = Bt(H If T = (C, {A 1,, A t } is the bloc tree of some graph, let X(T be the set of graphs i X with bloc tree T, ad dee U(T similarly Let B(T be the bipartite subgraph of B iduced o X(T U(T, ad let us estimate U(T / X(T If H U(T ad e [] ( \ E(H is a edge coectig A i ad A j with i j, the the bloc tree of H = H {e} diers from T Hece, the degree of H i this bipartite graph is at most u T = m + 1 A i A j + t 1 1 i<j t Note that the term t 1 correspods to the umber of edges i C Now let G X(T ad e E(G We have Bt(G e = T if ad oly if e G[A i ] \ R(G[A i ] for some i [t] Hece, the degree of G i B(T is t x T (G = R(G[A i ] (t 1 Suppose that t 3 or mi{ A 1, A } > 1 The by Lemma 10, we have x T (G u T = 1 i<j t A i A j (t 1 Settig x T = u T +, we have x T (G x T Boudig the edges of B(T i two dieret ways, we get t R(G[A i ] X(T x T e(b(t U(T u T We ow cosider the remaiig case, whe t = ad mi{ A 1, A } = 1 Note that we eed ot cosider the case t = 1 as T is ot the bloc tree of a -edge-coected graph Without loss of geerality, let A 1 = 1 We have u T M (, while x T (G M R(G[A ] for every G X(T Let a be the umber of graphs G i X(T with R(G[A ] r + 3 By Lemma 9, we have (( ( 1 (r+3/ a < + ɛr/, if > 1 (ɛ/ Coutig the umber of edges of B(T two ways, we get the followig bouds: Hece, ( X(T a(m (r + e(b(t (M ( U(T U(T M (r + ( r (4 4r = 1 + > 1 + X(T a M ( M ( ( 1 If X(T > 3 U(T a, this implies X(T r 1 Let T 0 be the set of pairs T = (C, {A 1, A } satisfyig the followig coditios: T is the bloc tree of some graph i C, A 1 = 1, ad X(T 3 a Let X 0 = T T 0 X(T Note that T 0 < as we have at most choices for A 1 ad at most 1 choices for the oe edge i C Hece, we have X 0 5 a This gives the followig boud o the size of (X 13

14 ( 1 + ( (X r 1 Therefore, if X 4 9( ( ( (r+1/ +ɛr/ 4 4r ( X X 0 1 (( ( (r+1/ X 4 5, the + ɛr/ (X X r But if is sucietly large give ɛ, we have (( ( (r+1/ ɛr/ < (( ( (r+1/ + ɛr X I the ext lemma, we show that if the umber of -edge-coected graphs i X is ot i the same rage as the umber of o--edge-coected graphs i X, the X < (X Lemma 19 There exists 5 such that if > 5, the followig holds Let M + 1 < M +, X C ( ad Y = X D, Z = X Y Suppose that Z > Y or Y > Z The (X > X If X ( m / (, we are doe by Lemma 16 So we ca suppose that X < m / Firstly, cosider the case whe Y > Z Let Y = ( y, the y < m / As (Y = (Y, we ca apply Lemma 15 to get (Y Y > m / 1 + Hece, (X (Y > Y + Y / > Y + Z Now cosider the case whe Z > Y Let Z = ( z, the z = /4 + O( Set ɛ = 1/40 ad r = /3 We choose r ad ɛ such that r < ad z > ( (r+1/ + ɛr holds Hece, by Lemma 18, we have (Z Z r/ , for sucietly large Estimatig the size of the shadow of X with (Z, we get (X (Z Z + 4 Z 3 Z + Y = X We also eed the followig techical lemma, which tells us what coditios eed to be satised for the sizes of the shadows of Y, Z to have X < (X 14

15 Lemma 0 Let a, b, c 1, c, c 3 be positive real umbers ad A = a(1 + c 1, B = b(1 + c ad C = a(1 c 3 If c 3 c 1 c, the a + b C + max{b, A C} We eed to show that ac 3 + b < max{b, A C} Observe that we ca suppose that B = A C Otherwise, if B < A C, we ca substitute b with b > b,ad B with B = b (1 + c 3, satisfyig B = A C The the left had side of the iequality icreases, while the right had side does ot chage We ca proceed similarly if A C < B If B = A C, the b = c 1+c 3 1+c a Hece, our iequality becomes ac 3 + c 1 + c c a (c 1 + c 3 a Simplifyig this iequality, we get that it is equivalet with c 3 c 1 c Now we are ready to show the existece of a complete matchig betwee the levels close to the middle level Theorem 1 There exists 6 such that if > 6, the followig holds If M + 1 < M +, the there exists a complete matchig from C ( to C ( 1 By Hall's theorem, it is eough to prove that for ay X C (, we have X (X Fix ɛ = 1/18 Let X = ( ( x By Lemma 16, if x > 1 + ɛ, the we are doe if > (ɛ Now suppose that x ( ( 1 + ɛ Let Y = X D ad Z = X Y Let Y = y (, Z = z, ad suppose that > 5 By Lemma 19, if Y > Z or Z > Y, we are doe Hece, we ca suppose that x ɛ < y, z x, if is sucietly large Let U = (Y D ad let r be a positive iteger satisfyig ( ( (r + 1/ r/ + ɛ(r + 1 x < + ɛr Oe ca easily chec that as x < ( 1 + ɛ ad ɛ < 1/4, such a r always exists, it is uique, ad r < Furthermore, y, z > ( (r+1/ + ɛr By Lemma 17, if > 3 (ɛ, we have Also, by Lemma 15 (Y Y U Y > 1 4r y + 1 > ( r/, + ɛr where the term ɛr comes from boudig 1 + ɛr above by ɛr Usig that > m/, we have = ( r/ > + ɛr ( r/ m/ = + ɛr m/ 1 1 r( 1/( 1 + r /( 1 + 8ɛr/( 1 > 15

16 Figure : The comparability graph betwee X ad its shadow > 1 1 r/ + r / + 9ɛr/, where the last iequality holds if is sucietly large Fially, by Lemma 18, if > 4 (ɛ, we have (Z Z Now we are ready to estimate (X We have where deotes disjoit uio Hece, > r/ (X = U (( (Y \ U (Z, (X U + max{ (Y U, Z } Also, X = Y + Z Let c 1 = r/ r / 9ɛr/ 1 r/+r / +9ɛ, c = 4 4r/ ad c 3 = 4r/ We have (Y > (1 + c 1 Y, (Z > (1 + c Z ad U > (1 c 3 Y Hece, by Lemma 0, our tas is reduced to provig that c 3 c 1 c Namely, 4r Simplifyig this iequality, we get r/ r / 9ɛr/ 1 r/ + r / + 9ɛr/ 4 4r/ 1 r/ + r / + 9ɛr/ ( r/ 9ɛ(1 r/ For our coveiece, let α = r/ The the previous iequality ca be writte as which reduces to 1 α + α / + 9ɛα ( α/ 9ɛ(1 α, α + 18ɛ As α < 1 ad ɛ = 1/18, this iequality holds Hece, if is sucietly large, we have X (X We are ow ready to prove our mai theorem Proof of Theorem Let > 6, where 6 is the costat give i Lemma 1 By Lemma 11, it is eough to prove that for = 1,, M 1 there is a complete matchig from C ( to C (+1, ad 16

17 for = M + 1,, m, there is a complete matchig from C ( to C ( 1 But we proved exactly this statemet i Lemma 13, Lemma 14 ad Theorem 1 As a al remar, we observe that the proof also shows that C (M is the uique largest atichai, as the strict iequality (X > X holds 5 Ope problems I this sectio, we propose several ope problems The rst problem we propose is ispired by the questio ivestigated i [4], which we metioed i the Itroductio Let G be a coected graph ad let C (G be the family of subgraphs of G that are coected o the vertex set V (G Dee the partial orderig < o C (G as usual: H < H if E(H E(H Questio Let G be a coected graph Is (C (G, < Sperer? We believe that there should be graphs G for which (C (G, < is ot Sperer Ufortuately, eve for small graphs, it is dicult to chec this property We also propose aother variatio of Questio 1 Let GP be a mootoe graph property (a family of graphs closed uder isomorphism, ad addig edges ad let GP deote the family of graphs i GP with vertex set [] Also, for = 0,, ( ( let GP be the set of graphs i GP with edges Dee the partial orderig < o GP as usual The poset (GP, < might ot be graded, however it still maes sese to as the followig questio For which graph properties GP is it true that the largest atichai i (GP, < is GP ( for some? To as a more specic questio, we propose the followig problem Questio 3 Let H be the family of Hamiltoia graphs Is (H, < Sperer? Fially, we suggest the followig variatio of Questio 1 Suppose we do ot distiguish graphs that are isomorphic More precisely, dee the equivalece relatio o C such that G H if G ad H are isomorphic, ad let C 0 be the set of equivalece classes of C Dee < o C 0 such that for G, H C 0 we have G < H if there exists G G ad H H satisfyig G < H i (C, < Questio 4 Is (C 0, < Sperer? 6 Acowledgemets We would lie to tha the aoymous referees for their useful commets ad suggestios, ad Adrew Thomaso for drawig our attetio to the simple proof preseted i Claim 8 Refereces [1] I Aderso, Combiatorics of Fiite Sets, Oxford Uiversity Press (1987 [] Flajolet, Sedgewic, Aalytic Combiatorics, Cambridge Uiversity Press (009 [3] P Hall, O Represetatives of Subsets, J Lodo Math Soc, 10 (1 (1935: 6-30 [4] M S Jacobso, A E Kézdy, S Seif, The poset o coected iduced subgraphs of a graph eed ot be Sperer, Order, 1 (3 (1995:

18 [5] Gy O H Katoa, Persoal commuicatio [6] Gy O H Katoa, A theorem of ite sets, Theory of Graphs, Aadémia Kiadó, Budapest (1968: [7] J B Krusal, The umber of simplicies i a complex, Mathematical Optimizatio Techiques, Uiv of Califoria Press (1963: [8] L Lovász, Combiatorial Problems ad Exercises, North-Hollad, Amsterdam (1993 [9] E Sperer, "Ei Satz über Utermege eier edliche Mege", Mathematische Zeitschrift (i Germa, 7 (1 (198: