Maximising the number of independent sets in connected graphs

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1 Maximisig the umber of idepedet sets i coected graphs Floria Leher ad Stepha Wager September 30, 015 A Turá coected graph TC,α is obtaied from α cliques of size α or α by joiig all cliques by a edge to oe cetral vertex i oe of the larger cliques. The graph TC,α was show recetly by Bruyère ad Mélot to maximise the umber of idepedet sets amog coected graphs of order ad idepedece umber α. We prove a geeralisatio of this result by showig that TC,α i fact maximises the umber of idepedet sets of ay fixed cardiality β α. Several results (both old ad ew o the umber of idepedet sets or maximum idepedet sets follow as corollaries. 1 Itroductio Turá s theorem [18], which characterises the K r -free graphs with greatest umber of edges, is probably the most classical result i extremal graph theory. It has bee modified ad exteded i various ways: oe such extesio, due to Erdős [6, 7] (ad rediscovered by Sauer [17] ad Roma [15], states that the complete (r 1-partite graph whose partite sets are as equal i size as possible (this graph is ofte called a Turá graph, but we will rather use this ame for its complemet later eve has the greatest umber of copies of K k amog all K r -free graphs of give order for every k {1,,..., r 1} (ad thus also the greatest total umber of iduced complete subgraphs; for k > 1, it is uique with this property. See also [1, Chapter VI, Corollary 1.10]. A similar result, due to Hedma [10], states that the same type of graph also has the greatest umber of cliques (maximal complete subgraphs amog all graphs of order with clique umber ω if ω <. We will be workig with the dual settig, cosiderig idepedet (stable sets rather tha complete subgraphs. The umber of idepedet sets, kow also as the Fiboacci Departmet of Mathematics, Uiversity of Hamburg, Budesstraße 55, 0146 Hamburg; mail@ floria-leher.et Departmet of Mathematical Scieces, Stellebosch Uiversity, Private Bag X1, Matielad 760, South Africa; swager@su.ac.za. The work of this author was supported by the Natioal Research Foudatio of South Africa uder grat umber

2 umber (i view of the fact that the umber of idepedet sets of a path is a Fiboacci umber, see [14] for the first referece or Merrifield-Simmos idex (a ame that stems from the work of chemists Merrifield ad Simmos [13] of a graph, has received quite a lot of attetio recetly, with a specific focus o extremal problems, see the survey [19] for a collectio of results. It is very atural to ask the followig questio: amog all graphs of order whose idepedece umber is α, which graphs have the greatest umber of idepedet sets? Of course, the aswer to this questio follows trivially from the aforemetioed theorem of Erdős if o further assumptios are made, sice idepedet sets of a graph correspod to complete subgraphs i the complemet. We ote, however, that the complemet of a complete multipartite graph is ot coected; i a recet paper, Bruyère ad Mélot [5] address the atural questio how the situatio chages if oly coected graphs are cosidered. They prove that the aswer i this case is give by what they call Turá-coected graphs. These graphs are obtaied from a uio of α cliques whose sizes are as equal as possible (the complemet of the aforemetioed complete multipartite graphs by addig additioal edges goig out from a vertex i oe of the large cliques to reder the graph coected. For trees, Bruyère, Joret ad Mélot cosidered the aalogous miimisatio problem (least umber of idepedet sets i [4] ad obtaied a partial characterisatio of the extremal trees. The mai result of this paper is to show that a more geeral statemet, parallelig the result of Erdős, holds: the Turá-coected graph has the greatest umber of idepedet sets of ay cardiality amog coected graphs of give order ad idepedece umber. Apart from the theorem of Bruyère ad Mélot, this has other implicatios as well: i particular, the Turá-coected graph maximises the umber of maximum idepedet sets amog coected graphs of give order ad idepedece umber. Comparig the umber of maximum idepedet sets for differet idepedece umbers, we fid the greatest possible umber of maximum idepedet sets a graph of order ca have, ad characterise the correspodig extremal graphs. This result was origially obtaied by Jou ad Chag [11], based o work of Griggs, Gristead ad Guichard [9] ad Füredi [8]. Oe also otes that the Turá-coected graph is a tree wheever α (there are o trees whose idepedece umber is lower. Therefore, we obtai the solutio to aother problem, which was solved by Zito [0] (ad geeralised by Saga ad Vatter [16], as a special case: which tree of order has the greatest umber of maximum idepedet sets? This paper is orgaised as follows: we first review some importat otatio ad prelimiaries, the formally state ad prove the mai result i Sectio 3. Our approach mostly follows [5], adapted to the more geeral settig. Several corollaries coclude the paper. Notios ad prelimiary results As usual, let α(g be the idepedece umber of G, that is, α(g is the greatest cardiality of a idepedet set i G. Deote by i β (G the umber of idepedet sets

3 of size β i the graph G. Throughout this paper we say that a graph is extremal for β if it maximises i β amog all coected graphs with the same idepedece umber. Defiitio.1. Let, α N. The Turá graph T,α o vertices is the disjoit uio of α cliques of sizes α ad α. Note that this defies T,α up to isomorphism because there is oly oe way to obtai vertices. Defiitio.. The Turá coected graph TC,α o vertices is obtaied from T,α by takig a vertex c i a clique of size α ad coectig it via a edge to exactly oe vertex i each of the other cliques. The vertex c is called the cetral vertex or cetre of the Turá coected graph, the clique cotaiig c is called the cetral clique. The followig recursio is stadard, yet very useful: Propositio.3. For ay vertex v of G, we have i β (G = i β 1 (G N[v] + i β (G v, where N[v] deotes the closed eighbourhood of v. Proof. Every idepedet set of size β either cotais v (ad thus oe of its eighbours or it does ot. Theorem.4 (see [15, Theorem 1]. Let α, β, be itegers with 1 < β α. For every graph G o vertices with idepedece umber α, the iequality i β (G i β (T,α holds, with equality oly if G is isomorphic to T,α. Lemma.5. If α, β 1 are fixed, the the value of i β (T,α is strictly icreasig i. Proof. Note that T +1,α ca be obtaied from T,α by addig a vertex v to oe of the smaller cliques. I the larger graph we have all idepedet sets from the smaller graph ad some idepedet sets obtaied by swappig i the ew vertex v for oe of the vertices i the clique that v was added to, which proves that i β (T +1,α > i β (T,α. Call a edge e of G critical for α if α(g e > α(g. A graph is called critical if all of its edges are critical. It is kow that a critical graph caot have a cutset spaig a complete subgraph, thus i particular o cut vertex: Lemma.6 (cf. [5, Lemma 1]. If G is critical ad coected, the G is -coected, i.e. G v is coected for every vertex v. For a proof, see [1, Problem 8.18]. If we remove edges from a graph, the umber of idepedet sets (of ay fixed cardiality β caot decrease. Startig from ay extremal graph, we ca thus remove edges util we reach a edge miimal graph, where removal of aother edge either icreases the idepedece umber or reders the graph discoected. It is clear that if a edge 3

4 miimal extremal (i particular coected graph is ot critical, the every ucritical edge must be a bridge. Furthermore, the followig lemma states that i this case we ca fid a ucritical bridge such that oe compoet of G e is critical, ad every edge cotaied i this compoet is also a critical edge of G (we call this a critical decompositio. It is essetially idetical to [5, Lemma 11], but sice our defiitio of extremality is differet, we provide its proof for completeess. Lemma.7 (cf. [5, Lemma 11]. A edge miimal coected graph that is extremal for some β is either critical, or it has a critical decompositio. Proof. Let G be a edge miimal coected graph, ad suppose that G is ot critical. Ay edge of G that is ot critical has to be a bridge, for otherwise we could remove it to obtai a coected graph of the same order ad idepedece umber, cotradictig edge miimality. Amog all ucritical edges, pick a edge e for which the smaller of the two compoets of G e, which we call G 1, has the smallest possible umber of vertices. Note that G 1 caot cotai ay ucritical bridge e of G, sice the smaller compoet of G e would be strictly smaller tha (ad i fact cotaied i G 1, cotradictig the choice of e. Thus all edges of G 1 are critical edges of G. We claim that G 1 itself is critical as well, which meas that we have foud the desired critical decompositio. If this is ot the case, the G 1 has a ocritical edge f, i.e. α(g 1 f = α(g 1. Now if G is the other compoet of G e, the we clearly have α(g f α(g e f = α(g 1 f + α(g = α(g 1 + α(g = α(g e = α(g, where the last idetity is due to the fact that e is ucritical by assumptio. This implies that f is ucritical for G as well, which is a cotradictio, completig the proof. Remark.8. As ca be see from the proof, the result of Lemma.7 holds for all edge miimal graphs, it does ot deped o extremality with respect to i β. 3 Mai result Theorem 3.1. Let α, β, be itegers with 1 β α <. For every coected graph G o vertices with idepedece umber α, we have i β (G i β (TC,α. For β >, the graph TC,α is the oly edge miimal extremal graph. Before we give the proof (i several steps, let us state ad prove the followig corollary: Corollary 3.. For < β mi{α, α} the graph TC,α is the oly extremal graph. Proof. It suffices to prove that addig a edge to TC,α strictly decreases the umber of idepedet sets of size β. I other words, we must show that ay two vertices that are ot coected by a edge are cotaied i a idepedet set of size β. 4

5 Pick two arbitrary o-adjacet vertices u, v. Of the α cliques that form TC,α, there are at least mi{α, α} whose size is at least. Sice at most two of them ca be fully cotaied i N[u] N[v], we ca fid β cliques which are ot cotaied i this set. From each of these cliques select a vertex to obtai a idepedet set of size β cotaiig both u ad v. Remark 3.3. For β = 1, evidetly all graphs are extremal. The extremal graphs for β = were characterised by Bougard ad Joret i []: they are obtaied from either a Turá graph or a graph whose compoets are odd cycles ad/or sigle edges by addig the miimum umber of edges (possibly zero required to obtai a coected graph. If α > ad β > α, there are also other extremal graphs besides the Turá coected graph: they are obtaied from TC,α by addig additioal edges icidet with the cetral vertex. The proof of Theorem 3.1 is iductive. It is easy to see that the theorem holds for 3. From ow o we will assume that we have show the theorem for all graphs o less tha vertices. Lemma 3.4. Complete graphs are extremal if ad oly if α = 1. Proof. They are the oly graphs with α = 1. Lemma 3.5. Odd cycles are extremal if ad oly if α = 1 ad β =. Proof. The idepedece umber of a odd cycle is α = 1, hece we oly cosider graphs with this idepedece umber. Clearly beig extremal for β = is equivalet to havig the smallest possible umber of edges. The oly coected graphs with fewer edges tha cycles are trees. However, every tree with a odd umber of vertices has a idepedet set of cardiality at least +1 (the larger part of its bipartitio. Thus a graph with idepedece umber α = 1 caot have more idepedet -sets tha the cycle. Coversely assume β >. I this case we will show that TC, 1 cotais more idepedet sets of size β tha C. Observe that the graph TC, 1 cosists of a triagle with 3 paths of legth attached to oe of its vertices. Choosig v i Propositio.3 as oe of the other two vertices of the triagle we obtai i β (TC, 1 O the other had we have = i β (TC 1, 1 + i β 1 (T 3, 3. i β (C = i β (P 1 + i β 1 (P 3. By our geeral iductio hypothesis i β (P 1 i β (TC 1, 1, ad by Theorem.4 i β 1 (P 3 < i β 1 (T 3, 3. Hece C is ot extremal. Next we characterise all critical extremal graphs, parallelig part of the proof of Theorem 1 i [5]. This also fixes a small flaw of the proof i [5], where a upper boud for the umber of idepedet sets of TC,α is used i oe case rather tha a lower boud. 5

6 Lemma 3.6. The oly extremal critical graphs for β are complete graphs ad odd cycles. Proof. Assume that we have a critical graph G which is either complete or a odd cycle. We will show that i this case G caot be extremal. Brooks theorem [3] tells us that the chromatic umber χ of G is less or equal to the maximum degree. Usig the trivial lower boud α for the chromatic umber we obtai which immediately implies that χ α, 1 α 1. Let v be a vertex of G with maximum degree. coected by Lemma.6. Hece we have Sice G is critical, G v must be i β (G = i β (G v + i β 1 (G N[v] i β (TC 1,α + i β 1 (T 1,α 1 i β (TC 1,α + i β 1 (T. α 1,α 1 If 1 mod α, there is exactly oe clique of size α i TC,α. I this case, applyig Propositio.3 to a vertex i this clique other tha the cetral vertex gives us i β (TC,α = i β (TC 1,α + i β 1 (T α,α 1 > i β (TC 1,α + i β 1 (T, α 1,α 1 the iequality beig a cosequece of Lemma.5. Otherwise, we choose a clique of size α that does ot cotai the cetral vertex ad apply Propositio.3 to a vertex i this clique that is ot adjacet to the cetral vertex. This gives us i β (TC,α = i β (TC 1,α + i β 1 (TC α,α 1 i β (TC 1,α + i β 1 (T, α 1,α 1 where the iequality is due to fact that T α 1,α 1 is a proper subgraph of TC α,α 1 (it is obtaied by removig the cetral vertex. I either case, we ed up with i β (G i β (TC,α. Hece for G beig extremal it is ecessary that all the iequalities hold with equality. I the first case ( 1 mod α, we have already see that this is impossible. I the secod case, we must have i β 1 (T α 1,α 1 ( = i β 1 TC, α,α 1 6

7 which ca oly happe if there is o idepedet set of cardiality β 1 i TC α,α 1 that cotais the cetral vertex, or equivaletly o idepedet set of cardiality β i TC,α that cotais the cetral vertex. This happes if ad oly if α β + 1 (cf. the proof of Corollary 3.. But the α α + β + 1, so α =. Now Lemma.5 tells us that the iequality i β 1 (T 1,α 1 i β 1 (T α 1,α 1 ca oly be sharp if = α =. Sice G is coected, this meas that G is either a path or a cycle. Eve cycles are ot critical, ad either are paths with more tha two vertices, so this completes the proof. Before we start the proof of our mai theorem, we eed oe last defiitio: Defiitio 3.7. Call a graph almost Turá coected if there is a vertex v (the cetral vertex such that G v is the disjoit uio of cliques, there is a clique (the cetral clique, possibly empty such that v is adjacet to each of its vertices, ad for every other clique there is exactly oe edge coectig it to v. I other words, it has the same structure as a Turá coected graph, but the sizes of the cliques are ot ecessarily balaced. Proof of Theorem 3.1. For β = 1, every graph is extremal, ad the case β = is settled by [, Propositio 5] (cf. Remark 3.3. For the rest of this proof assume that β 3 ad that G is a edge-miimal extremal graph for β. By Lemma.7 we kow that either G is critical, or there is a critical decompositio of G. I the first case Lemma 3.6 provides a full aswer: G is complete, sice it caot be a odd cycle for β 3 i view of Lemma 3.5. Hece cosider the case where there is a critical decompositio of G. The iductio step i this case cosists of four parts: (i show that the critical compoet of a critical decompositio must be extremal as well, (ii show that there is a critical decompositio where the critical compoet is complete, (iii show that G is almost Turá coected, ad (iv show that it is best possible to have balaced clique sizes. For the first step, assume that we have a critical decompositio of G ito parts G 1 ad G ; we will deote their orders by 1 ad ad their idepedece umbers by α 1 ad α respectively. Without loss of geerality assume that G 1 is the critical compoet. Let e = uv be the bridge coectig G 1 ad G, with u G 1 ad v G. Sice e is a ucritical bridge, the idepedece umber of G is α = α 1 + α. Propositio.3 gives i β (G = γ 0 ( iγ (G 1 i β γ (G v + i γ (G 1 ui β γ 1 (G N[v]. (1 7

8 Note that this sum cotais oly fiitely may o-zero summads ad that at least oe of the summads must be o-zero uless β > α. By the iductio hypothesis, Furthermore we have α(g 1 u α 1 ad hece i γ (G 1 i γ (TC 1,α 1. ( i γ (G 1 u i γ (T 1 1,α 1. (3 Note here that G 1 u eed ot be coected. Summig up yields i β (G β ( iγ (TC 1,α 1 i β γ (G v + i γ (T 1 1,α 1 i β γ 1 (G N[v]. γ=0 Equality holds for every γ if G 1 is the Turá coected graph TC 1,α 1, u beig the cetral vertex. Thus equality i ( must hold for every γ α 1 where i β γ (G v 0 as we assumed G to be extremal. If β α 1, we ca take γ = β, for which we trivially have i 0 (G v = 1 0. Thus G 1 is extremal for β i this case. Otherwise, we take γ = α 1. Sice the idepedece umber of G v is at least α 1 = α α 1 1, we have i β γ (G v 0 uless β γ α, which is oly possible if β = α. So G 1 is extremal for α 1, except perhaps for the case that β = α ad i α (G v = 0. I that case, every maximum idepedet set of G (ad thus also every maximum idepedet set of G cotais v. I coclusio, either G 1 is extremal for mi{β, α 1 }, or every maximum idepedet set of G cotais v (ad hece does ot cotai u. I the latter case, ay edge icidet to u i G 1 is ucritical for G ad hece G 1 is either a sigleto (which already proves (i ad (ii or it cotais a ucritical edge, which is a cotradictio. Thus the proof of (i is complete. Note that we oly used the criticality of G 1 at the ed. So by the same argumets as above we ca show that G either is extremal for mi{β, α }, or every maximum idepedet set of G cotais u ad does ot cotai v. We will use this fact later. For step (ii assume that either G 1 or G is complete, otherwise there is othig to show. I particular, α 1 ad α. We already kow that G 1 is extremal for γ = mi{α 1, β} ad thus it must be a odd cycle. There is a maximum idepedet set i G 1 that does ot cotai the vertex u, so G is extremal for mi{β, α } by the remark at the ed of step (i. Sice β 3 we ca use the iductio hypothesis to show that either G is a Turá coected graph or α =. If G is a Turá coected graph, the it cotais a ucritical bridge that we ca remove to obtai a critical decompositio with a complete critical compoet. If G is ot a Turá coected graph the α =, ad sice G is extremal for α as well, G is a C 5. So the oly remaiig problematic case is whe G cosists of a 5-cycle ad aother odd cycle coected by a sigle edge. I this sceario, we have α = 1 (recall that is the umber of vertices, so the correspodig Turá coected graph TC,α = TC, 1 has two cliques of three vertices, all other cliques are of size. Apply Propositio.3 to 8

9 a arbitrary vertex x of G that is ot adjacet to either edpoit of the bridge, ad to a vertex of the 3-vertex clique i TC 1, 1 that does ot cotai the cetral vertex, to obtai the followig iequality: i β (G = i β (G x + i β 1 (G N[x] < i β (TC 1, 1 + i β 1 (TC 3, = i β (TC, 1 This shows that G is ot extremal, completig the proof of (ii. I the followig, we ca assume that G 1 is complete. For the proof of (iii oce agai recall that G must be extremal for γ = mi{β, α } uless every idepedet set of size β i G uses the vertex u. Sice G 1 is complete it is easy to see that i the latter case G 1 ca oly be a sigleto, so that (1 reduces to i β (G = i β (G + i β 1 (G v i β (TC 1,α + i β 1 (T,α. The oly way that equality ca hold here if G is ot isomorphic to TC,a is that β > α, so α = β 1. However, G v must still be isomorphic to T 1,α. Every compoet of G v must be coected to v by at least oe edge i G. Sice G is edge miimal, we coclude that there is exactly oe edge from each compoet to v ad hece G is almost Turá coected i this case. Hece we may assume that G is extremal for mi{β, α }. If this miimum is 1 the α = 1 ad G is a clique. If the miimum is the α = ad we ca argue as before: either G is a C 5, or it is Turá coected. The former possibility ca be ruled out directly agai: i this case, α = β = 3, i 3 (G = 5 7 (recall that G 1 is complete, ad mod 3, i 3 (TC,3 = mod 3, mod 3, so it is easily verified that i 3 (G < i 3 (TC,3. Thus G is Turá coected if mi{β, α } =, ad if mi{β, α } 3 this is also the case by the iductio hypothesis. For the proof of (iii it oly remais to show that it is best possible to have v at the cetre c of the Turá coected graph. Assume that v c ad let G be the graph obtaied from G by replacig e = uv by e = uc. Every idepedet set of G which does ot cotai u is also a idepedet set of G ad vice versa. Similarly the idepedet sets which do cotai u but either v or c are the same for G ad G. Hece we oly have to compare the umber of idepedet sets of G cotaiig u ad c to the umber of idepedet sets i G cotaiig u ad v. To ivestigate those umbers, observe that G (N[u] N[c] ad G (N[u] N[v] are both disjoit uios of cliques, ad that the cliques of G (N[u] N[v] are eve balaced, i.e., it is a Turá graph. Thus ( i β (G (N[u] N[c] i β T α α +1,α 1 9

10 with equality if ad oly if v is a eighbour of c, ad ( i β (G (N[u] N[v] i β T 1,α α 1 with equality if ad oly if v is cotaied i a o-cetral clique of size β 3 ad α 3, Lemma.5 tells us that α. Sice i β (G (N[u] N[c] < i β (G (N[u] N[v], ad hece G caot be extremal. This cotradictio completes the proof of (iii. So we kow that G is almost Turá coected. It oly remais to show that balaced clique sizes are best possible ad that the cetral clique must be oe of the larger cliques. For this purpose let k be the size of some clique C ad let l be the size of the cetral clique C 0. If k = l or k = l 1 there is othig to show. We ow cosider the cases k > l ad k < l 1 ad show that i both cases G is ot extremal. This will be doe by movig a vertex from a larger clique to a smaller clique, i.e., discoectig this vertex from all of its eighbours ad coectig it to every vertex of the smaller clique. It is readily verified that the umber of idepedet sets of size β cotaiig a vertex i at most oe of the two ivolved cliques is ivariat uder this trasformatio. For coveiece, we simply deote this umber by A β. We will show that the umber of idepedet sets of size β cotaiig vertices i both cliques icreases, so that we obtai a graph with the same idepedece umber ad more idepedet sets of size β. I the case where k > l we move a vertex from C to C 0 to obtai a ew graph G. Before movig the vertex we have i β (G = (k 1I β + k(l 1I β + A β, where I β deotes the umber of idepedet sets of size β i the remaiig cliques ad I β deotes the umber of such cliques which do ot cotai vertices adjacet to the cetre. After movig the vertex we have i β (G = (k I β + (k 1lI β + A β. The differece of the two values is ow easily see to be positive: i β (G i β (G = I β + ki β li β > I β + ki β li β = (k l 1I β 0 because k > l ad both k ad l are itegers. Hece i β (G > i β (G. Sice the idepedece umbers of G ad G coicide this cotradicts G beig extremal. 10

11 Now assume that k < l 1. I this case we move a vertex from C 0 to C to obtai a ew graph G. We have Takig differeces gives i β (G = ki β + (k + 1(l I β + A β. i β (G i β (G = I β + (l k I β > 0 by similar argumets as above. But this agai cotradicts G beig extremal. Hece we have show that if G is edge miimal but ot critical, the it must be a Turá coected graph. Together with Lemma 3.6 this completes the proof of the theorem. 4 Cosequeces As it has already bee metioed i the itroductio, our mai result has a umber of implicatios. First ad foremost, it obviously implies the result of Bruyère ad Mélot that was the origial motivatio for this paper: Corollary 4.1 (cf. [5, Theorem 1]. The Turá-coected graph TC,α has the greatest umber of idepedet sets amog all coected graphs of order whose idepedece umber is α. Sice Theorem 3.1 holds i particular for β = α (i.e., the umber of idepedet sets whose cardiality is equal to the idepedece umber, we also have the followig corollary: Corollary 4.. The Turá-coected graph TC,α has the greatest umber of maximum idepedet sets amog all coected graphs of order whose idepedece umber is α. Ay tree of order has idepedece umber at least. We otice that for ay α, the Turá-coected graph TC,α is ideed a tree. Therefore, we also immediately obtai the followig results: Corollary 4.3 (cf. [5, Corollary 13]. For ay α, the Turá-coected graph TC,α has the greatest umber of idepedet sets amog all trees of order whose idepedece umber is α. Corollary 4.4. For ay α, the Turá-coected graph TC,α has the greatest umber of maximum idepedet sets amog all trees of order whose idepedece umber is α. Comparig the umbers for differet values of α, we also fid the extremal coected graphs or trees without restrictios o the idepedece umber. Let us first cosider arbitrary coected graphs. Griggs, Gristead ad Guichard [9] ad idepedetly by Füredi [8] for sufficietly large determied the coected graphs of order with the 11

12 greatest umber of maximal idepedet sets (maximal with respect to set iclusio rather tha cardiality, but sice every maximum idepedet set is ecessarily maximal idepedet as well, ad coversely all maximal idepedet sets i the extremal graphs are i fact also maximum idepedet sets, the result remais true for maximum idepedet sets, as poited out by Jou ad Chag i [11]: Corollary 4.5 (cf. [11, Theorem 17]. A coected graph of order 5 has at most maximum idepedet sets, with equality for the complete graph or (if = 5 the Turá coected graph TC 5,. For 6, the uique coected graph of order with the greatest umber of maximum idepedet sets is the Turá coected graph TC, 3. Proof. For 5, the statemet ca be verified directly, so assume that 6. I view of Corollary 4.4, the maximum must be attaied by a Turá coected graph TC,α for some α. Cosider such a Turá coected graph, assume that it has the greatest umber of maximum idepedet sets, ad let l be the umber of vertices i the cetral clique. There are r = αl small cliques of size l 1 ad α r 1 large o-cetral cliques of size l, so we obtai the followig formula for the umber i α (TC,α of maximum idepedet sets of TC,α : i α (TC,α = l α r 1 (l 1 r+1 + (l 1 α r 1 (l r. If l 5, we replace the cetral clique by a clique of l vertices ad add aother clique of size that is joied to the cetral vertex by a edge. The the umber of maximum idepedet vertices chages to (l 3 l α r 1 (l 1 r + (l 1 α r 1 (l r i α (TC,α, with equality oly if l = 5. I the latter case, however, the resultig graph is ot Turá coected (sice 6. Thus we ca rule out the possibility that l 5. O the other had, if l =, the we replace two o-cetral cliques by a sigle oe (two -cliques become oe 4-clique, a -clique ad a sigleto become a 3-clique, or two sigletos become a -clique. Agai, the umber of maximum idepedet sets does ot decrease, ad the resultig graph is ot Turá coected, so we reach a cotradictio. Thus l = 3 or l = 4. Now we ote that for l = 3, the fuctio 3 α r 1 r+1 + α r 1 = 3 α 1 3α +1 + α 1 is decreasig i α, while for l = 4, the fuctio 4 α r 1 3 r α r 1 r = 4 3α 1 3 4α α 1 4α is icreasig i α. This leaves us with the followig optios: if = 3s, i s (TC 3s,s = 3 s 1 + s 1 or i s 1 (TC 3s,s 1 = 16 3 s s 4, if = 3s + 1, i s+1 (TC 3s+1,s+1 = 8 3 s + s or i s (TC 3s+1,s = 3 s + s 1, if = 3s +, i s+1 (TC 3s+,s+1 = 4 3 s 1 + s 1 or i s (TC 3s+,s = 4 3 s s. 1

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