INDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE

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1 INDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE JAMES ALEXANDER, JONATHAN CUTLER, AND TIM MINK Absrac The enumeraion of independen ses in graphs wih various resricions has been a opic of much ineres of lae Le i(g be he number of independen ses in a graph G and le i (G be he number of independen ses in G of size Kahn used enropy o show ha if G is an r-regular biparie graph wih n verices, hen i(g i(k r,r n/2r Zhao used biparie double covers o exend his bound o general r-regular graphs Galvin proved ha if G is a graph wih (G and n large enough, hen i(g i(k,n In his paper, we prove ha if G is a biparie graph on n verices wih (G where n 2, hen i (G i (K,n when 3 We noe ha his resul canno be exended o = 2 (and is rivial for = 0, Also, we use Kahn s enropy argumen and Zhao s exension o prove ha if G is a graph wih n verices, (G, and (G, hen i(g i(k, n/2 Inroducion The sudy of independen ses in various classes of graphs has been a opic of much recen ineres For a graph G, we le I(G be he se of independen ses of G and i(g = I(G Kahn [8] made a breakhrough on hese problems when he proved he following resul wih a beauiful enropy argumen Theorem (Kahn If G is an r-regular biparie graph on n verices wih r, hen i(g i(k r,r n/2r = ( 2 r+ n/2r In fac, Kahn proved a sronger resul for weighed independen ses Galvin and Teali [5] generalized Theorem o homomorphisms and, in he process, exended Kahn s weighed independen ses resul Zhao [2] exended Theorem o all r-regular graphs using he biparie double cover of a graph Given a graph G, define he biparie double cover of G, denoed G K 2, o be he graph wih verex se V (G {0, } wih (u, i (v, if and only if uv E(G and i The key lemma of Zhao [2] was he following Lemma 2 (Zhao If G is any graph, hen wih equaliy if and only if G is biparie i(g 2 i(g K 2, The problem of maximizing he number of independen ses among graphs in oher classes has also been well-sudied If we consider graphs on n verices and m edges, hen he answer follows from he Kruskal-Kaona heorem [0, 9] Define he lex graph wih n verices and m edges, denoed L(n, m, o be he graph wih verex se [n] and edge se consising of an iniial segmen of size m of ( [n] 2 under he lexicographic ordering (Recall ha for ses A, B Z, we say A < B in he lex order if min(a B A The following is a consequence of he Kruskal-Kaona heorem (See [2] for an alernaive proof Corollary 3 If G is a graph wih n verices and m edges, hen i(g i(l(n, m The research of he firs auhor was suppored by The Margare and Herman Sokol Graduae Summer Research Fellowship

2 2 J ALEXANDER, J CUTLER, AND T MINK Maximizing he number of independen ses in several oher classes of graphs was considered in [3] The main focus of his paper will be on independen ses in graphs wih n verices and minimum degree The asympoic version of his problem was sudied by Sapozhenko [] in biparie graphs wih large minimum degree Recenly, Galvin [4] proved he following asympoic resul Theorem 4 (Galvin Fix > 0 There is a n( such ha for all n n(, he unique graph wih he mos independen ses is K,n In he same paper, Galvin conecured he following Conecure (Galvin If G is a graph on n verices wih minimum degree a leas, where n and saisfy n 2, hen i(g i(k,n One of he main resuls of his paper is a level ses version of his conecure for biparie graphs We le I (G = {I I(G : I = } and i (G = I (G For any graph G on n verices, we have ha i 0 (G = and i (G = n, so he problems of maximizing i 0 (G and i (G are no ineresing The problem of maximizing i 2 (G is a bi more ineresing Noe ha i 2 (G = ( n 2 e(g, and so maximizing i 2 (G corresponds o minimizing e(g Thus, if we are ineresed in maximizing i 2 (G where G is a graph wih n verices and minimum degree, we simply wan o make G as regular as possible Thus, i is no he case in general ha i 2 (G i 2 (K,n for G a graph wih n verices and minimum degree Galvin [4] was able o prove ha K,n is he unique maximizer for i (G wih 3 among graphs wih minimum degree one We believe ha i (G i (K,n for 3 and are able o prove as much when G is biparie Theorem 5 Le n,, and be posiive inegers wih n 2 and 3 If G is a biparie graph on n verices and minimum degree a leas, hen wih equaliy if and only if G = K,n i (G i (K,n, Even among biparie graphs i is, in general, no he case ha K,n maximizes he number of independen ses of size wo The cases when n = 2 or n = 2 + are no of much ineres since here is a unique biparie graph (K, or K,+ saisfying he condiions of he heorem If n 2 + 2, we know ha K +,n has fewer edges (and hus more independen ses of size wo han K,n In Secion 2, we will prove Theorem 5 along wih some relaed resuls For example, we can prove ha Theorem 5 implies he biparie case of Conecure In a relaed quesion, one migh hope o ge a bound on he number of independen ses in a graph in erms of is minimum and maximum degrees The following conecure was made by Kahn; see [6] We le iso(g be he number of isolaed verices in a graph G Conecure 2 (Kahn If G is any graph, hen ( i(g 2 iso(g 2 d(u + 2 d(v uv E(G d(ud(v In Secion 3, we modify he enropy proof of Kahn [8] o ge he following Theorem 6 If G is a biparie graph wih bipariion (A, B such ha (G, hen i(g ( / d(v From his, i is easy o derive a general bound using Zhao s exension

3 INDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE 3 Corollary 7 If G is a graph on n verices wih (G and (G, hen i(g i(k, n/2 Proof If H is any biparie graph wih minimum degree and (G hen, by applying Theorem 6 since one of is wo pars conains a mos half of is verices, we have i(h (2 + 2 n/2 = i(k, n/2 Thus, o prove he same bound for general G, we can use is biparie double cover (which also has he minimum and maximum degree of G and apply Lemma 2 o ge which yields he resul i(g 2 i(g K 2 i(k, 2n/2, While his resul does give a bound for non-regular graphs, i does no seem o be sharp excep when G is regular (and so reduces o Theorem In paricular, i would nice o ge a bound ha would imply Conecure in general We conecure he following Conecure 3 If G is a graph on n verices wih minimum degree a leas and maximum degree a mos, hen i(g i(k, + n We are able o prove a slighly sronger resul in he case when = and do so a he end of Secion 3 Theorem 8 If n and are inegers wih n and q and r are defined o be he unique inegers such ha n = q( + + r and 0 r < +, hen for any graph G on n verices wih (G and (G, i is he case ha i(g i(k, q i(k,r 2 Proof of Theorem 5 and relaed resuls In his secion, we begin by proving Theorem 5 Theorem 5 Le n,, and be posiive inegers wih n 2 and 3 If G is a biparie graph on n verices and minimum degree a leas, hen wih equaliy if and only if G = K,n i (G i (K,n, Proof Le G be any n-verex biparie graph of minimum degree a leas wih bipariion G = A B We may assume, wihou loss of generaliy, ha A B We know ha A, for if no, he verices of B could no saisfy he minimum degree requiremen Define he ineger c so ha A = + c Thus, B = n c and 0 c n2 2 since A B We know ha independen ses in G can be pariioned ino hose conained enirely in A, hose conained enirely in B, and hose conaining verices from boh A and B Le Λ = {I I (G : I A, I B } So, as A and B are hemselves independen ses, we have ( ( ( ( A B + c n c i (G = + + Λ = + + Λ Our firs goal will be o bound Λ To his end, noe ha Λ = {I I (G : I B = } = {(b, I : I I (G, I B =, b I} b B

4 4 J ALEXANDER, J CUTLER, AND T MINK If I I (G is such ha I B = and b I for some b B, hen he verices of I A canno be in he neighborhood of b and so, since d(b, here are a mos A = c verices in A from which o choose he verices of I A Furher, he verices of (I B \ {b} mus no be in any of he neighborhoods of he verices in I A This oin neighborhood mus have size a leas and so here are a mos n 2 c verices lef o choose he verices of (I B \ {b} Thus, we have Λ = = {(b, I : I I (G, I B =, b I} b B ( n 2 c b B n c = n c n 2 c ( n 2 c ( n 2 c ( We will now use his bound on Λ o show ha if 3, hen he difference i (K,n i (G is nonnegaive Using (, we see ( n i (K,n i (G = =0 + ( ( n c n c n 2 c ( n c ( ( c n c = = + ( ( + c ( n 2 c ( n c =0 n c ( ( c n 2 c n 2 c ( n c ( ( c n 2 c n 2 c [( n c ( n c n 2 c ( n 2 c ( ],

5 INDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE 5 where he firs equaliy follows from Vandermonde s ideniy Expanding and applying his again o he above, we have ( [ c ( ( ( ] n 2 c n 2 c i (K,n i (G = l l n 2 c l= ( ( [ ( ] c c n 2 c = + (n 2 c 2 n 2 c 2 ( [ c ( ( ( ] n 2 c n 2 c +, (2 l l n 2 c =3 where he second equaliy holds because 3 We hen noe ha l= ( ( n 2 c l l l= ( n 2 c > n 2 c So, o show ha i (K,n i (G 0, i suffices o show ha ( [ n 2 c (n 2 c 2 n 2 c 2 ] ( n 2 c ( c 0 (3 Applying Vandermonde s ideniy once more, we have ha he lef hand side of he above is equal o ( [ ( ] ( [ ] c n 2 c + c ( (c + = c (c + 2 Furher, since 3, we have 2(c + 2 2(c ( (c +, and so he above expression is nonnegaive To show ha K,n is he unique maximizer of i (G, we noe ha in (3, he inequaliy is sric and so i (G > i (K,n unless c = 0 This, in urn, implies ha G = K,n since he only biparie graph wih A = saisfying he minimum degree condiion is K,n Remark We noe ha in he case = 2, he equaion (2 would become [ i 2 (K,n i 2 (G c (n c n c ] (n 2 c = c, (4 n 2 c which makes sense, for any graph wih less edges han K,n should have more independen ses of size 2 In paricular, e(k,n e(g (n (n c = c for any G wih minimum degree a leas Theorem 5 has several implicaions, some of which we presen here To begin, we prove ha i implies Conecure for biparie graphs In fac, i gives a bound on he independence polynomial for biparie graphs wih given minimum degree For a graph G, we le he independence polynomial of G, denoed P (G, x, be he generaing funcion for independen ses in G, ie, P (G, x = α(g =0 i (Gx, where α(g is he independence number of G We are able o show, as a corollary of he proof of Theorem 5, ha K,n maximizes P (G, x for all x Porism 2 If G is an n-verex biparie graph wih minimum degree a leas where n 2, hen P (G, x P (K,n, x for all x wih equaliy if and only if G = K,n

6 6 J ALEXANDER, J CUTLER, AND T MINK Proof Using he firs wo erms on he righ hand side of (2 and also he bound in (4, we see ha P (K,n P (G, x = (i (K,n i (Gx =0 [ ( ( ] c c n 2 c + cx x 2 2 =3 ( c = x + n 2 c + ( c x 2 2 =2 =3 ( c = x + n 2 c + ( c x + 2 =2 =2 ( = + n 2 c + x x 2 =2 If c = 0, hen = 0, each erm in he sum is zero If c, hen each coefficien is a leas ( c ( + c+ n2 2 x as c 2, which is clearly non-negaive when c, x Leing x =, he following is immediae Corollary 22 Suppose n and are posiive inegers wih n 2 If G is an n-verex biparie graph wih minimum degree a leas, hen i(g i(k,n wih equaliy if and only if G = K,n The nex resul of his secion proves he level se version of Conecure when n = 2 Theorem 23 If G is any 2-verex graph wih minimum degree a leas, hen i (G i (K,n = i (K, for all 0 Proof We show his by inducion on We have ha i (G = i (K, rivially Assume ha i (G i (K, Le J (G = {(v, I : v I, I I (G} Then we have So, ( + i + (G = J + (G (n i (G i + (G n i (G n i (K, = i + (K,, + + where he second inequaliy is by inducion and he las sep follows from n = 2 The following corollary immediaely follows Corollary 24 If G is any 2-verex graph wih minimum degree a leas, hen P (G, x P (K,, x for any x 0 3 A differen bound In his secion, we prove a weak version of Conecure 3 based on Kahn s enropy proof of Theorem In fac, we use ideas from Galvin and Teali s exension of Theorem o general homomorphisms [5] The proof uses enropy mehods and so we begin his secion wih a reminder of some basic facs abou enropy A more deailed inroducion can be found in, for example, [7] Throughou his secion, all logarihms are base wo Definiion The enropy of a random variable X is defined by H(X = P(X = x log P(X = x x

7 INDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE 7 For random variables X and Y, he condiional enropy of X given Y is defined by H(X Y = E(H(X Y = y = y P(Y = yh(x Y = y Enropy has some naural, and useful, properies, some of which we include in he following heorem Theorem 3 ( If X is a random variable, hen H(X log range(x wih equaliy if and only if X is uniform on is range (2 If X = (X, X 2,, X n is a random sequence, hen H(X = H(X + H(X 2 X + + H(X n X, X 2,, X n (3 If X, Y, and Z are random variables, hen H(X Y, Z H(X Y Par (2 of he heorem is usually called he chain rule, and we will call par ( he uniform bound and par (3 he informaion loss bound We also make use of he following lemma, known as Shearer s Lemma [] If X = (X, X 2,, X n is a random sequence and A [n], we wrie X A for he random sequence (X a a A Theorem 32 (Shearer Le X = (X, X 2,, X n be a random sequence and A be a collecion of subses of [n] such ha each elemen i [n] is in a leas k elemens of A Then H(X H(X A k A A Wih he preliminaries of enropy ou of he way, we will resae and prove he main heorem of his secion Theorem 6 If G is a biparie graph wih bipariion (A, B such ha (G, hen i(g ( / d(v Proof Le G = (V, E be a biparie graph wih bipariion (A, B wih A B, so ha A n/2 Choose an independen se I uniformly from I(G and define a random vecor X = (X v v V where X v = if v I and X v = 0 if v I Since I is chosen uniformly, we know ha H(X = log i(g We have H(X = H(X B + H(X A X B [ ] H(X N(v + H(X v X N(v = [ H(XN(v + H(X v X N(v ], where he firs equaliy is he chain rule and he firs inequaliy is by he informaion loss bound and Shearer s lemma wih A = {N(v : v A} We hen noe ha if here is a verex w N(v such ha X w =, hen X v mus be 0 We le Q v = {X w : w N(v} and, for R {0, }, q v (R = P(Q v = R Also, for R {0, }, le s v (R be he number of R-labelings of he verices in N v in which all elemens of R are used (ie, he number of surecions from N v o R and v (R

8 8 J ALEXANDER, J CUTLER, AND T MINK be he number of possible values of X v given ha Q v = R Thus, s v ({0} = s v ({} = and s v ({0, } = 2 d(v 2 We also have ha v ({0, } = v ({} = and v ({0} = 2 We hen see ha [ H(XN(v + H(X v X N(v ] = [ H(Qv + H(X N(v Q v + H(X v X N(v ] =R {0,} =R {0,} [ q v (R log q v (R + q v (RH(X N(v Q v = R ] + q v (RH(X v X N(v, Q v = R [ q v (R log q v (R ] + q v (R log s v (R + q v (R log v (R (5 = log =R {0,} q v (R log s v(r v (R q v (R s v (R v (R (6 =R {0,} = ( log d(v, where we used he uniform bound on enropy repeaedly, he definiion of condiional enropy for (5, and Jensen s inequaliy for (6 Remark 2 We noe ha his argumen can be generalized o homomorphisms ino any image graph us as Galvin and Teali proved [5] This gives an upper bound on he number of homomorphisms o any image graph from a graph wih given minimum and maximum degree We conclude he paper by proving a slighly sronger version of he = case of Conecure 3 Theorem 8 If n and are inegers wih n and q and r are defined o be he unique inegers such ha n = q( + + r and 0 r < +, hen for any graph G on n verices wih (G and (G, i is he case ha i(g i(k, q i(k,r Proof Le G be a graph wih minimum degree a leas one and maximum degree a mos Form a graph G by removing edges from G unil every edge is inciden o a verex of degree one Noe ha i(g i(g and also ha G mus be he disoin union of sars Tha is, k G = K,ni, i=

9 INDEPENDENT SETS IN GRAPHS WITH GIVEN MINIMUM DEGREE 9 where k + i n i = n Suppose ha G has wo componens ha are no K, s, ie, here are s < k such ha n s, n < Assume ha n s n Noe ha i(k,x = 2 x +, so ha i(k,ns i(k,n = (2 ns + (2 n + < (2 ns + (2 n+ + = i(k,nsi(k,n+ Repeaing his process, we see ha here is a mos one componen in an exremal graph ha is no K, Thus, we have i(g i(g i(k, q i(k,r References F R K Chung, R L Graham, P Frankl, and J B Shearer, Some inersecion heorems for ordered ses and graphs, J Combin Theory Ser A 43 (986, no, Jonahan Culer and A J Radcliffe, Exremal graphs for homomorphisms, J Graph Theory 67 (20, no 4, , Exremal problems for independen se enumeraion, Elecronic J Combin 8 (20, no, #P69 4 David Galvin, Two problems on independen ses in graphs, Discree Mah 3 (20, no 20, David Galvin and Prasad Teali, On weighed graph homomorphisms, Graphs, morphisms and saisical physics, DIMACS Ser Discree Mah Theore Compu Sci, vol 63, Amer Mah Soc, Providence, RI, 2004, pp David Galvin and Yufei Zhao, The number of independen ses in a graph wih small maximum degree, Graphs Combin 27 (20, no 2, Charles M Goldie and Richard G E Pinch, Communicaion heory, London Mahemaical Sociey Suden Texs, vol 20, Cambridge Universiy Press, Cambridge, 99 8 Jeff Kahn, An enropy approach o he hard-core model on biparie graphs, Combin Probab Compu 0 (200, no 3, G Kaona, A heorem of finie ses, Theory of graphs (Proc Colloq, Tihany, 966, Academic Press, New York, 968, pp Joseph B Kruskal, The number of simplices in a complex, Mahemaical opimizaion echniques, Univ of California Press, Berkeley, Calif, 963, pp AA Sapozhenko, On he number of independen ses in biparie graphs wih large minimum degree, DIMACS Technical Repor (2000, no Yufei Zhao, The number of independen ses in a regular graph, Combin Probab Compu 9 (200, no 2, address, J Alexander: alexander@mailmonclairedu address, J Culer: onahanculer@monclairedu address, T Mink: mink@mailmonclairedu (J Alexander, J Culer, T Mink Deparmen of Mahemaical Sciences, Monclair Sae Universiy, Monclair, NJ 07043