Technische Universität Ilmenau Institut für Mathematik

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1 Technische Universität Ilmenau Institut für Mathematik Preprint No. M 07/14 Locally dense independent sets in regular graphs of large girth öring, Frank; Harant, Jochen; Rautenbach, Dieter; Schiermeyer, Ingo 2007 Impressum: Hrsg.: Leiter des Instituts für Mathematik Weimarer Straße Ilmenau Tel.: Fax: ISSN xxxx-xxxx

2 Locally Dense Independent Sets in Regular raphs of Large irth Frank öring 1 Jochen Harant 2 Dieter Rautenbach 2 and Ingo Schiermeyer 3 1 Fakultät für Mathematik, TU Chemnitz, D Chemnitz, ermany, frank.goering@mathematik.tu-chemnitz.de 2 Institut für Mathematik, TU Ilmenau, Postfach , D Ilmenau, ermany, jochen.harant@tu-ilmenau.de, dieter.rautenbach@tu-ilmenau.de 3 Institut für Diskrete Mathematik und Algebra, Technische Universität Bergakademie Freiberg, Freiberg, ermany, schierme@math.tu-freiberg.de Abstract. For an integer d 3 let αd be the supremum over all α with the property that for every ɛ > 0 there exists some gɛ such that every d-regular graph of order n and girth at least gɛ has an independent set of cardinality at least α ɛn. Extending an approach proposed by Lauer and Wormald Large independent sets in regular graphs of large girth, J. Comb. Theory, Ser. B , and improving results due to Shearer A note on the independence number of triangle-free graphs, II, J. Comb. Theory, Ser. B , and Lauer and Wormald, we present the best known lower bounds for αd for all d 3. Keywords. regular graph; independence; girth; randomized algorithm 1

3 1 Introduction In the present paper we consider the independence number α of finite, simple and undirected graphs = V, E which are d-regular for some d 3 and have large girth. For integers d 3 and g 3 let d, g denote the class of all d-regular graphs of girth at least g and let αd, g := sup {α α α V for all = V, E d, g}. Clearly, αd, g is monotonic non-decreasing in g and bounded above by 1 and we can consider αd := lim g αd, g. Note that this definition implies that for every ɛ > 0 there exists some gɛ such that α αd ɛ V for every graph = V, E d, gɛ. Our aim is to prove lower bounds on αd. While the first result on the independence number in regular graphs of large girth is due to Hopkins and Staton [5] who proved α , for quite a long time the 18 best known estimates of αd were due to Shearer [8]. Theorem 1 Shearer 1991 If then for all d 3. β Shearer d := for d = 3 and 1+dd 1β Shearer d 1 d 2 +1 for d 4, αd β Shearer d Only very recently Lauer and Wormald [6] improved Shearer s result for d 7. Theorem 2 Lauer and Wormald 2007 For all d 3 αd β LauWo d := 1 d 1 2/d 2. 2 From a very abstract viewpoint their approaches are actually similar. On the one hand Shearer constructs an independent set by carefully selecting vertices according to some degree dependent weight function, adding them to the independent set, deleting them together with their neighbours and iterating this process. On the other hand Lauer and Wormald construct an independent set by randomly selecting vertices, adding most of them to the independent set, deleting them together with their neighbours and iterating this process. In order to get some intuition about how to improve these approaches it is instructive to see that a very simple argument allows to improve Shearer s bound on α3. 2

4 Proposition 3 α > β Shearer Proof: It follows from Theorem 4 in [8] that for every ɛ > 0 there is some gɛ such that: If = V, E is a graph of order n and girth at least gɛ, with n 2 vertices of degree 2 no two of which are adjacent and n 3 = n n 2 vertices of degree 3, then α ɛ n ɛ n 3. 1 For a cubic graph of order n and sufficiently large girth gɛ the 9-th power 9 is = 1533-regular. Therefore, 9 has an independent set I 9 with I 9 n/ = n/1534 where 9 denotes the maximum degree of 9 cf. e.g. [4, 9]. Let H arise from by deleting all vertices within distance at most 3 from I 9. We construct an independent set of by adding all 7 I 9 many vertices at distance 0 or 2 from a vertex in I 9 and by applying 1 to H. It follows that α 7 I 9 + αh 79 7 I ɛ 24 I ɛn which completes the proof ɛ n I 9 The proof of Proposition 3 suggests that it is worthwhile to consider the iterative deletion not just of a vertex and its neighbours which would induce a rooted tree of depth 1 but of rooted trees of larger depths. Locally this should allow us to pack the vertices of the independent set more densely which hopefully yields an overall improvement. We follow exactly this intuition by generalizing the random procedure and its analysis using differential equations proposed by Lauer and Wormald in [6]. 2 The Algorithm T REEk, l, p, f In this section we describe a random procedure T REEk, l, p, f which depends on two integers k, l 0, a real value p [0, 1] and a function f which maps rooted trees T with root r to independent subsets ft, r of their vertex set. We assume that ft, r = ft, r for isomorphic rooted trees, T rooted at r and T rooted at r. The algorithm T REEk, l, p, f will be applied to a graph = V, E of girth at least 2l + 1. It executes k rounds and determines disjoint rooted subtrees of of depth at most l. The value p will serve as a probability. We need a little more notation to describe the algorithm. Let u V be a vertex of the graph. For an integer i 0 let N i u and N u denote the sets of vertices of 3

5 within distance measured with respect to exactly i from u and at most i from u, respectively, i.e. Nu i = {v V dist v, u = i} and N i u = {v V dist v, u i}. Furthermore, let B i u denote the set of vertices v N u which are not adjacent to a vertex in V \ N i u, i.e. Bu i = { v V N 1 i v N u}. For a set U V of vertices of the subgraph of nduced by V \ U is denoted by U or [V \ U]. T REEk, l, p, f proceeds as follows: 1 Set 0 = V 0, E 0 :=, Z 0 := and i := 0. 2 While i < k select a subset X i of V by assigning every vertex of to X i independently at random with probability p. Set 3 Output Z k. +1 = V i+1, E i+1 := u X i V i N l u, 2 X i := {v X i dist v, u 2l + 1 u X i \ {v}}, 3 T i u := [ B l i u ], 4 Z i := u X i V i f T i u, u, 5 Z i+1 := Z i Z i, 6 i := i + 1. There are some subtleties we want to stress: The definition of +1 in 2 and Z i in 5 use neighbourhoods within the graph while X i is defined in 3 with respect to. Furthermore, the construction of Z i+1 in 4, 5 and 6 does not influence the evolution of the. By the girth condition, T i v is a tree and f T i v, v is a well-defined subset of B l v. We first observe that T REEk, l, p, f really produces an independent set of. Lemma 4 Z k is an independent set of. 4

6 Proof: For contradiction, we assume that v, w Z k with vw E. Let v Z i and w Z j with, say, i j. Let v f T i u, u for some u Xi V i. Since, by the definition of Xi in 3, the set N l u N l u is empty for all distinct u, u Xi we obtain that w V i is a neighbour of v outside of N l u which implies the contradiction v B l i u. 3 The Analysis of T REEk, l, p, f Throughout this section we will assume that = V, E is a d-regular graph for some d 3 and sufficiently large girth. We consider the behaviour of T REEk, l, p, f when applied to this graph. We will specify the necessary girth conditions which are all in terms of k and l more exactly whenever they are explicitely needed. It is one of the key observations made by Lauer and Wormald in [6] that for a sufficiently large girth the probabilities which are suitable to describe the behaviour of their randomized algorithm can be well understood. The next lemma corresponds to Lemma 2 in [6]. Lemma 5 Let k 2 and 0 i k. Let the girth of be at least 2k + 1l + 2 and let u V and vv E. i The probabilities P[u V i ], P[v V i v V i ] and P[u Z i ] as well as the conditional expected value E [ f T i u, u u X i V i ] do not depend on the choice of the vertex u or the edge vv. ii Conditional upon the event v V i, the event v V i depends only on the intersection of the sets X 0, X 1,..., X i 1 with N il v v. Proof: It follows immediately, by induction on i, from the description of T REEk, l, p, f that the events u V i and v V i v V i depend only on the intersection of the sets X 0, X 1,..., X i 1 with N il il il u and N v N v, respectively. Furthermore, the event u Z i depends only on the intersection of the sets X 0, X 1,..., X i 1 with N i+1l u and the intersection of the set X i with N 2l u. Finally, conditional upon the event u Xi V i, the cardinality of f T i u, u depends only on the intersection of the sets X 0, X 1,..., X i 1 with N i+1l u. [ [ ] Since, by the girth condition, the induced subgraphs N 2l ], u N i+1l u and [ ] N il il v N v are isomorphic for all choices of the vertex u or the edge vv, we obtain i. Similarly, ii follows immediately by induction on i. 5

7 By Lemma 5, for 0 i k the following quantities r i := P[u V i ], w i := P[v V i v V i ] P[v V i ] = P[v V i v V i v V i ], f l w i := E [ f T i u, u u X i V i ], z i := P[u Z i ] and z i := P[u Z k \ Z i ] are the same for every vertex u V and every edge vv E. Using Lemma 5, we can determine the following recursions for these probabilities. Lemma 6 Let the girth of be at least 2k + 1l + 2. i r 0 = w 0 = 1 and z i+1 = z i z i for 0 i k 1. ii For 0 i k 1 r i+1 = r i 1 p 1 + w i+1 = w i 1 p 1 + dd 1 j 1 w j i + Op 2, d 2d 1 j 1 w j i + Op 2 2l z i = f l w i r i p 1 p dd 1j 1 and where the constants implicit in the O -terms depend only on d and l. Proof: i is immediate from the definitions and we proceed to the proof of ii. Let u V be fixed. For v N l u let P v denote the vertex set of the unique path of length at most l from u to v. The event u V i+1 holds if and only if u V i u X i v X i v X i P v V i. v:1 dist u,v l Expanding this representation of the event u V i+1 to a disjunction of conjunctions, all events corresponding to the conjunctions are disjoint because they differ in X i {v : 1 dist u, v l}. 6

8 Furthermore, all of those events for which two of the independent events v X i for some v with 1 dist u, v l hold, will contribute together only P[u V i ] Op 2 to P[u V i+1 ] where the constant implicit in the O -term depends only on d and l. Therefore, P[u V i+1 ] = P u V i u X i v X i + P[u V i ] Op 2 + v:1 dist u,v l P v:1 dist u,v l u V i v X i P v V i v :v v 0 dist u,v l v X i = P[u V i ] 1 p 1+P l P l + P [u V i P v V i ] p 1 p dd 1j 1 v:1 dist u,v l +P[u V i ] Op 2 = P[u V i ] + 1 v:1 dist u,v l +P[u V i ] Op dd 1j 1 dd 1 j 1 p P [u V i P v V i ] p In order to evaluate P [u V i P v V i ] let u = u 0 u 1 u 2...u j = v be the unique path from u to v for some 1 j l. By Lemma 5 i and ii, we have for 0 ν j 1 P [u 0, u 1,..., u ν V i u ν+1 V i ] = P [u 0 V i ] P [u 1 V i u 0 V i ] P [u 2 V i u 0, u 1 V i ]... P [u ν V i u 0, u 1,..., u ν 1 V i ] P [u ν+1 V i u 0, u 1,..., u ν V i ] = P [u 0 V i ] P [u 1 V i u 0 V i ] P [u 2 V i u 1 V i ]... P [u ν V i u ν 1 V i ] 1 P [u ν+1 V i u ν V i ] = r i w ν i 1 w i 7

9 and we obtain P [u V i P v V i ] = P [u 0 V i u 0 V i u 1 V i... u j V i ] Putting everything together, we obtain r i+1 = P[u V i+1 ] = r i 1 = r i 1 p = r i 1 p 1 + = P [u 0 V i u 1 V i ] +P [u 0, u 1 V i u 2 V i ] +P [u 0, u 1, u 2 V i u 3 V i ]... +P [u 0, u 1,..., u j 1 V i u j V i ] = r i 1 w i + r i w i 1 w i r i w j 1 i 1 w i = r i 1 w j i. dd 1 j 1 p + dd 1 j 1 r i 1 w j i p + r i Op 2 dd 1 j 1 r i w j i p + r i Op dd 1 j 1 w j i + Op 2. By the same type of argument, it follows that for every edge vv E P[v V i+1 v V i+1 ] = P[v V i v V i ] 1 p 2 + 2d 2d 1 j 1 w j i + Op 2. Since w i = P[v V i v V i ] r i, the desired equation for w i follows. Finally, we consider z i. By the definitions in 3 and 5 and Lemma 5, we have z i = P[u Z i ] = E[ Z i ] V = f l w i P[u X i V i ] = f l w i P[u V i ] P[u X i ] which completes the proof. 2l = f l w i r i p 1 p dd 1j 1 8

10 Setting for 0 i k and a i := z iw i r i = P [u Z k \ Z i u V i ] w i w i r i := r i+1 r i, w i := w i+1 w i and a i := a i+1 a i for 0 i k 1, we obtain the following. Lemma 7 For 0 i k 1 d 1w i l 1 d 1w i 1 a i = f lw i 2a i w d 1w i 1 + d 2w i l 1 i d 1w i 1 + Op where the constant implicit in the O -term depends only on d and l. Proof: Note that, by definition, z i = z i z i+1. Immediately from the previous definitions it is straightforward to verify that a i = w w i r i i w a i r i+1 r i r i+1 i z i w i+1 r i. w i w i w i r i w i r i+1 By Lemma 6, w i+1 w i = 1 Op and r i r i+1 = 1 + Op. Furthermore, w i l 1 = 1 + d 2w i w i d 1 j p + Op 2, w i j=0 r i l 1 = 1 + dw i w i d 1 j p + Op 2 and r i j=0 z i r i = f l w i p + Op 2 which implies that also r i l 1 = 1 + dw i w i d 1 j p + Op 2. r i+1 j=0 9

11 Putting everything together, we obtain a i = f lw i 2a l 1 i j=0 d 1w i j + Op w i 1 + d 2w l 1 i j=0 d 1w i j + Op. Note that f l w i is bounded from above by the order of a d-regular tree of radius l, i.e. it is bounded above in terms of d and l. Clearly, 2a l 1 i j=0 d 1w i j is bounded from above in terms of d and l while 1 + d 2w l 1 i j=0 d 1w i j is bounded from below by 1 and bounded from above in terms of d and l. Altogether this implies the stated equation for a i w i. We proceed to our main result which extends Theorem 1 of [6]. Theorem 8 Let d 3 and l 0. If f l w is continuous on [0, 1], then αd b l,f 1 where b l,f is the solution of the linear differential equation with b l,fw = c l,f,0 w + c l,f,1 wb l,f w and b l,f 0 = 0 7 c l,f,0 w = f l w 1 + d 2w d 1wl 1 d 1w 1 and Proof: Note that by definition, c l,f,1 w = 2 d 1wl 1 d 1w d 2w d 1wl 1 d 1w 1 a 0 = z 0w 0 r 0 = z 0 = P[u Z k \ Z 0 u V 0 ] = P[u Z k ]. Therefore, T REEk, l, p, f produces an independent set of = V, E of expected cardinality a 0 V and hence, by the first moment principle, αd a 0. Whenever we use the O -notation, the implicit constants will be in terms of d and l. Similarly as in [6], we will prove that for every ɛ > 0 there is some c 0 = c 0 ɛ and a function p 0 c > 0 such that for pk = c > c 0 and p < p 0 c we have which clearly implies the desired result. a 0 b l,f 1 Oɛ. 10

12 Let some ɛ > 0 be fixed. By Lemma 6, we have w i+1 w i 1 p + O p 2 for 0 i < k. Therefore, for sufficiently small p, w k 1 p/2 k = 1 p/2 1 c p. 8 Thus for p small enough and c we have w k 0. Furthermore, by Lemma 6, and hence w i 0 as p 0 uniformly for every 0 i < k. w i = Op 9 Since f l w is continuous, the function c 0 w := c l,f,0 w is continuous and bounded on the compact set [0, 1]. Note that f l w is always bounded in terms of d and l as already noted in the proof of Lemma 7. Furthermore, the function c 1 w := c l,f,1 w is Lipschitz continuous on [0, 1]. Hence the solution bw := b l,f w of 7 is also Lipschitz continuous on [0, 1] where all bounds and Lipschitz constants are in terms of d and l. Clearly, a k = 0. Let bw be the solution of the differential equation with modified initial condition b w = c 0 w + c 1 w bw and bwk = 0. By the mentioned continuity/ Lipschitz continuity conditions, it follows from standard results cf. Corollary 4 and Corollary 6 in 7 of [2] that the solution of 7 depends continuously on the initial contition. Hence, by 8, for p small enough and c large enough, bwk = bw k + Oɛ which implies By Lemma 7, we have for 0 i < k that which implies b1 = b1 + Oɛ. a i = a i+1 a i = a i+1 c 0 w i + c 1 w i a i + Op w i a i = a i+1 c 0 w i + Op w i 1 + c 1 w i w i = a i+1 c 0 w i w i 1 + c 1 w i w i + Op w i 10 for p small enough. Similarly, the differential equation for b together with the mean value theorem imply for 0 i < k and some w i+1 w i w i that bwi = bw i+1 c 0 w i + c 1 w i b w i w i. 11

13 By 9 and the continuity of c 0, c 1 and b, this implies that for every δ > 0 there is some p 1 δ such that for p < p 1 δ bwi = bw i+1 c 0 w i + c 1 w i bw i + Oδ w i and thus for p small enough. In view of 10 and 11, we deduce Since for p small enough we obtain, by induction, bwi = bw i+1 c 0 w i + Oδ w i 1 + c 1 w i w i = bw i+1 c 0 w i w i 1 + c 1 w i w i + Oδ w i 11 bwk a k = 0 and for 0 i < k bwi a i = bw i+1 a i c 1 w i w i + Op + Oδ w i c 1 w i w i = 1 + O w i = 1 + Op, k 1 bw0 a 0 = Op + Oδ w i 1 + Op i i=0 We have 1 + Op k = 1 + Op 1 c p which is bounded in terms of c. Therefore, choosing δ small enough in terms of c and choosing p small enough in terms of c and δ, we finally obtain b1 = a 0 + b1 b1 + b1 a0 = a 0 + Oɛ and the proof is complete. 3.1 Some Instructive Choices for l It is very instructive to consider the behaviour of T REEk, l, p, f for l {0, 1} and appropriate choices for f. 12

14 For l = 0 we have N 0 v = {v} and Xi = X i for all 0 i k 1. Furthermore, the set B l i v contains the vertex v exactly if all neighbours of v in are not contained in V i and is empty otherwise. Choosing ft, r = V T whenever the vertex set V T of T satisfies V T 1 and ft, r = otherwise, T REEk, 0, p, f produces an independent set by Lemma 4. Conditional upon the event u Xi V i, the expected value of ft i u, u equals f 0 w i = 1 w i d, because, by Lemma 5 ii, each of the d neighbours of u are in V i independently at random with probability w i. The differential equation 7 simplifies to b 0,fw = f 0 w and b 0,f 0 = 0 which has the solution b 0,f w = 1 1 wd+1 1+d and thus b 0,f 1 = d. Therefore, by Theorem 8, asymptotically T REEk, 0, p, f produces an independent set of which contains exactly the same fraction of the vertices, namely 1, as guaranteed by 1+d the lower bound on the independence number of d-regular graphs proved by Caro [4] and Wei [9]. The reason for this is that for pk and p 0, T REEk, 0, p, f essentially processes the vertices of according to a random linear ordering v 1, v 2,..., v n and adds an individual vertex v i to the constructed independent set exactly if all neighbours of v i are among {v 1 v 2,..., v i 1 }. Applying this algorithm to a random linear ordering, the probability that an individual vertex v belongs to the constructed independent set equals exactly 1/1 + d, because this is the probability that v is the last vertex from N 1 v with respect to the linear ordering. In fact, this is exactly the argument used in [1] to prove the bound due to Caro and Wei [4, 9]. For l = 1 the set B 1 v always contains the vertex v itself and we can choose ft, r = {r} in order to obtain an independent set according to Lemma 4. For this choice T REEk, 1, p, f essentially coincides with the randomized p-greedy algorithm used by Lauer and Wormald in [6] for p = p 1, p 2,..., p k = p, p,..., p. Solving the differential equation 7 yields the same values for b 1,f 1 as obtained by Lauer and Wormald. In the next section, we consider a choice of f which generalizes their p-greedy algorithm. 3.2 A Reasonable Choice for f A reasonable choice for f is to select all vertices within some even distance from the root. Therefore, let l = 2h + 1 for some integer h 0 and let f even T, r = {u V T dist T r, u is even}. 13

15 Note that B 2h+1 u contains all vertices of N 2h+1 u within distance at most 2h from u. Conditional upon the event u Xi V i, the probability for the event v B 2h+1 u for some vertex v with dist u, v = 2j for some 1 j h equals exactly the probability that all vertices of the unique path from u to v within N 2h+1 u lie in V i. Using Lemma 5 in the same way as for the calculation of P [u 0, u 1,..., u ν V i u ν+1 V i ] in the proof of Lemma 6, this conditional probability equals w 2j i. By linearity of expectation, we deduce f even 2h+1 w = 1 + h dd 1 2j 1 w 2j h 1 = 1 + dd 1w 2 d 1w 2j j=0 = 1 + dd 1w 2 d 1w2h 1 d 1w 2 1. Now the differential equation 7 reads as follows. 1 + dd 1w 2 d 1w 2h 1 b d 1w 2h+1,f even w = 2 1 b 2h+1,feven 0 = d 2w d 1w2h+1 1 d 1w 1 2b 2h+1,feven w d 1w2h+1 1 d 1w 1 12 The algorithm proposed by Lauer and Wormald in [6] corresponds to the choice h = 0 in which case 12 simplifies to The solution of this differential equation is b 1,f even w = 1 2b 1,f even w 1 + d 2 w. b 1,feven w = wd 2 w 2/d 2 2 which together with Theorem 8 immediately implies one of the main results from [6]. Corollary 9 Lauer and Wormald, cf. Theorem 1 in [6] For every d 3, we have αd 1 d 1 2/d 2. 2 Next we consider the behaviour of b 2h+1,feven w for h. Our analysis naturally splits into the two cases d 1w < 1 and d 1w > 1. The intuitive reason for this is that for values of w i with d 1w i < 1 the sets N l u typically contain no vertices far from u, while for values of w i with d 1w i > 1 the sets 14

16 N l u may contain vertices up to distance l from u, i.e. the trees induced by the sets B l i u die out quickly for small values of the probability of survival w i. Considering the two intervals [ 1 0, d 1 and 1, 1] it follows from standard results cf. d 1 Corollary 6 in 7 of [2] that for h the solutions of 12 converge to the solutions of b,f even w = 1 + dd 1w2 2b,feven w d 1w 2 1 d 1w 1 dd 1w 2 d 1w d 2w d 1w 1 2b,feven w d 1w d 1w 1 d 2w d 1w d 1w 1 for 0 w < 1/d 1 for 1/d 1 < w 1 = b,feven 0 = 0. d 1w 2 +1 d 1w+11 w 2 1 w b,f even w for 0 w < 1/d 1 d d 1w+1d 2 2 d 2w b,f even w for 1/d 1 < w 1 13 Corollary 10 For every d 3, we have αd b,feven 1. Solving 13 for d = 3 yields w + w2 + 2 ln 2w w b,feven w = and hence in this case Similarly, we obtain for 0 w < 1/d 1 3 w ln2w + 1 ln2 for 1/d 1 < w 1 4 w w b,feven 1 = /8 ln An Optimal Choice for f ln for d = 4 b,feven for d = for d = 6. In this section we consider a function f opt for which f opt T, r is a maximum independent set within the rooted tree T. For some tree T with root r the set f opt T, r is obtained by applying the following algorithm A opt : Start with f opt T, r =. Iteratively add to f opt T, r all vertices at maximum distance from the root r within the current tree and delete them together with their parents. For this algorithm it follows immediately from the definition of B l i v that f opt T i v, v = B l v f opt [ 15 N l+1 ] v, v

17 for every v Xi V i. By Lemma 5, for 0 i < k and 1 j l + 1 the probability p l j, i = P [v f opt u, u u Xi V i N l+1 v N l+1 j ] u does not depend on the choice of u, v V with dist u, v = l + 1 j, i.e. p l j, i is well-defined. Furthermore, by Lemma 5 ii, the events v f opt and v f opt N l+1 N l+1 u, u u X i V i u, u u Xi V i are independent for different v, v N l+1 j u. v N l+1 j v N l+1 j Let t j w be defined recursively for integers j 1 by { 0 for j = 1 and t j w := 1 wt j 1 w d 1 for j 0. Obviously, by definition and the first step of the algorithm A opt, For j 0 the event v f opt N l+1 t 1 w i = p l 1, i = 0 and t 0 w i = p l 0, i = 1. u, u u Xi V i v N l+1 j u u u is equivalent to the event that none of the vertices v N 1 l+1 j 1 v N u is in the set f opt N l+1 u, u, i.e. either they are not in V i or they are in V i but not in the set f opt N l+1 u, u. By Lemma 5 ii, conditional upon the event u Xi V i v N l+1 j u, the probability that v N 1 l+1 j 1 v N u is not in V i equals 1 w i, and the probability that such a v is in V i but not in f opt N l+1 u, u equals w i 1 p l j 1, i. Hence, the probability that such a v is in not f opt N l+1 u, u equals and, by Lemma 5 ii, 1 w i + w i 1 p l j 1, i = 1 w i p l j 1, i N p l j, i = 1 w i p l j 1, i 1 l+1 j 1 v N u. 16

18 For 0 j l the values p l j, i satisfy the same recursion as the t j w i starting with the same value 0 at j = 1 and altogether we obtain { tj w p l j, i = i for 1 j l and 1 w i t l w i d for j = l + 1 for j = l + 1 remember that the root u has d possible children while all internal vertices have d 1. By linearity of expectation, f opt l w i = E [ fopt B l i u, u u X i V i ] = v N l u P = p l l + 1, i + = p l l + 1, i + [ v fopt B l u, u u X i V i ] dd 1 l j w l+1 j i p l j, i w i dd 1w i l j p l j, i = 1 w i t l w i d + w i dd 1w i l j t j w i l 1 = 1 w i t l w i d + w i d d 1w i j t l j w i. 14 In the case wd 1 < 1 and l the limit behaviour of the recursion for t j w becomes important. The function t 1 wt d 1 maps the unit interval [0, 1] into itself and the absolute value of its derivative d 1w1 wt d 2 is strictly smaller than 1 for t [0, 1] and wd 1 < 1. Hence the recursion for t j w is a contractive map and converges to a unique fixed point tw which solves the equation tw = 1 wtw d 1. Because for wd 1 < 1 the factors preceeding t j w in 14 decrease exponentially in j, we obtain lim f opt l w = 1 wtw d + twwd l = tw j=0 d 1w j j=0 wd 1 wtw + 1 d 1w 17

19 and Because in this case c,fopt,0w = lim l c l,fopt,0w = tw dw + 1 wtw1 d 1w. 1 w c,fopt,1w = lim l c l,fopt,1 = 2 w 1, 1 we are able to solve the differential equation for l in the interval [0, and obtain d 1 w b,fopt w = w 1 2 = w w 0 c,fopt,0w t 1 2 δt tw dw + 1 wtw1 d 1w 1 wt 1 2 δt. The integral is solveable at least for d = 3 in which case we obtain b,fopt w = 1 + 6w 4w w The most interesting value in this case is at = 1 d b,fopt = Clearly, f even w f opt w and we can use f even w in order to determine a lower bound for b,fopt 1 in the case d = 3 by solving 13 on the interval [ 1, 1] using as 2 initial condition the value b 1 [,fopt 2 at w = 1/2. We obtain for w 1, 1] the following 2 lower bound b,fopt w w + 12w ln w 16w 2. Corollary 11 α3 b,fopt ln > β Shearer 3. In general the following observations are useful for estimating lim l c l,fopt,0w in the case wd 1 > 1. 18

20 Observation 1 Because the recursion for t j w is based upon a strictly monotonic decreasing function which contracts the unit interval, and starts with t 1 w = 0 we obtain for j 0. t 2j w > t 2j+1 w, t 2j+1 w < t 2j w and t 2j+1+1 w > t 2j+1 w Observation 2 Consider a modified algorithm A applied to a tree T with root r which behaves like A opt up to some distance j + 1 from r, chooses less vertices at distance j from r than A opt and continues like A opt for smaller distances to r. For distances larger than j to r the output of A coincides with the output of A opt. For distances at most j to r the output of A coincides with the output of A opt when applied to a proper subtree of the tree induced by the vertices at distance at most j to r. Therefore, the set produced by A will contain at most as many vertices as the set produced by A opt. Conversely, if A + chooses more vertices at distance j from r than A opt possibly neglecting independence and behaves like A opt otherwise, then the set produced by A + will contain at least as many vertices as the set produced by A opt. Iteratively applying these observations allows to derive lower and upper bounds on f opt l w for d 1w > 1: We choose an integer j. If for all j j we replace in 14 t 2j 1 by t 2j 1 and t 2j by t 2j, then Observation 1 and Observation 2 for A imply that we obtain a lower bound on f opt l w. If for all j j we replace in 14 t 2j+1 by t 2j +1 and t 2j by t 2j, then Observation 1 and Observation 2 for A + imply that we obtain an upper bound on f opt l w. Therefore, we obtain f opt l w = 1 wt lw d wd 1 l wd 1 l wd 2j 2 + wd t j w wd l j j=j t j w wd 1 j t 2j w wd 1 2j + l+1 2 j=j t 2j 1w wd 1 2j 1 19

21 and f opt l w wd 1 l 1 wd 1 l +wd 2j 1 t j w wd l j j=j t 2j w wd 1 2j + l 1 2 j=j t 2j +1w wd 1 2j+1. Using these inequalities it is possible to derive lower and upper bounds for c,fopt,0w. and c,fopt,0w = c,fopt,0w = = wd 1 1 d 2w lim f opt l w l wd 1 l 2j 2 dwd 1 1 d 2 d d 2 dwd 1 1 d 2 d d 2 t j w wd 1 + j t 2j w + wd 1t 2j 1w 2j 2 wd 1 + 1wd 1 + 2j 2 2j 1 t j w wd 1 + j t 2j w + wd 1t 2j 1 wd 1 2j j=j t 2j w + wd 1t 2j +1w 2j 1 wd 1 + 1wd 1 + 2j 1 wd 1 1t j w wd 1 j wd 1t 2j +1w + t 2j w wd 1 2j+1 j=j wd 1 1t j w. wd 1 j Choosing j sufficiently large and numerically solving the corresponding two differential equations, we can obtain estimates for b,fopt 1 with any desired precision. Corollary 12 For d 3 we have αd b,fopt 1. The next table summarizes the numerically obtained values for selected values of d. The entry γd is an upper bound on αd which is derived from the analysis of random d-regular graphs [3, 7]. d max{β Shearer d, β LauWo d} b,fopt 1 γd

22 References [1] N. Alon and J. Spencer, The Probabilistic Method, Wiley, New York [2] V.I. Arnol d, Ordinary differential equations, Transl. from the 3rd Russian ed., Springer Textbook. Berlin etc.: Springer-Verlag. 334 p [3] B. Bollobás, The independence ratio of regular graphs, Proc. Amer. Math. Soc , [4] Y. Caro, New results on the independence number, Technical Report, Tel-Aviv University [5]. Hopkins and W. Staton, irth and independence ratio, Canad. Math. Bull , [6] J. Lauer and N. Wormald, Large independent sets in regular graphs of large girth, J. Comb. Theory, Ser. B , [7] B.D. McKay, Independent sets in regular graphs of high girth, Ars Combinatoria 23A 1987, [8] J.B. Shearer, A note on the independence number of triangle-free graphs, II, J. Comb. Theory, Ser. B , [9] V.K. Wei, A lower bound on the stability number of a simple graph, Bell Laboratories Technical Memorandum, , Murray Hill, NJ, [10] N. Wormald, Differential equations for random processes and random graphs, Annals of Applied Probability ,