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12. Probability Colin McDiarmid What are ‘typical’ properties of a graph? How can random methods yield existence results for graphs? What will randomised graph algorithms do? What have isoperimetric inequalities for graphs to do with probability? This chapter gives an introductory taste of the fertile area of graphs and probability. 1 Introduction Since Paul Erdős brought together the subjects of graph theory and probability some forty years ago, the relationship has flowered marvellously. Here we attempt to give something of the flavour of this still burgeoning area – no attempt is made at a complete survey! We discuss first some typical properties of random graphs and of algorithms on random graphs; for example, what can we say about the chromatic number and about methods to find good colourings? Then we see how to use random graphs in order to obtain deterministic results, and continue along this line with the help of the Lovász local lemma. The remaining two sections concern deterministic graphs. First we consider randomised algorithms on graphs – that is, procedures that operate on a deterministic graph and involve flipping coins. Finally we discuss the relationship between results on concentration of measure and isoperimetric inequalities for graphs, focussing on the n-cube Qn . 2 Random graphs: usual behaviour There are several natural models of random graph: we ³ ´shall focus on two of these. The random graph Gn,p has vertices v1 , . . . , vn , and the n2 possible edges occur independently, each with probability p. When p = 12 , this corresponds to sampling uniformly from all labelled graphs on n vertices. The random graph Gn,m also has vertices v1 , . . . , vn : each such graph with exactly³m´ edges is equally likely. Its behaviour is very similar to that of Gn,p , with p = m/ n2 . When we investigate random graphs, sometimes it is because we want to know whether a property usually holds, or how well an algorithm usually works, and some- 1 times our aim is to prove a deterministic result. In this section we discuss the usual behaviour. We shall focus on four topics: colourings, Hamiltonian cycles, minimum spanning trees and perfect graphs. For references not cited here, and for many further results, see [2]. When considering colourings we shall use the model Gn,p with the edgeprobability p constant (not depending on n). Vertex colouring The natural place to start is with the independence (or stability) number α(Gn,p ). We might guess that it should usually be close to r, where r is such that the expected ³ ´ r n number of stable r-sets, r (1 − p)(2) , is about 1. Such an r = r(n) is asymptotically equal to 2 log n/ log(1/(1 − p)), which equals 2 log2 n when p = 21 . The following result shows that α is remarkably predictable. Theorem 2.1 The independence number α satisfies α(Gn,p ) = r − 1 or r a.s. Here ‘a.s.’ (almost surely) means that the statement holds with probability that tends to 1 as n → ∞. The upper bound in the theorem is easy: it follows directly from the ‘first moment method’ – that, if always X ≥ 0, then Pr(X ≥ 1) is at most the expected value EX. The lower bound follows after some computations from the ‘second moment method’ – that if EX ≥ 0, then Pr(X > 0) ≥ (EX)2 /E(X 2 ), or a similar inequality. Here the random variable X counts the number of stable k-sets in Gn,p , where we take k as r + 1 or r − 1. The chromatic number χ(G) is at least the number of vertices divided by the stability number α(G). Hence, by Theorem 2.1, χ(Gn,p ) ≥ n/r a.s. It was conjectured in 1975 that this bound is about the correct value, and in 1988 this was proved by Bollobás [4]. In fact, we have the following rather precise result [21]: Theorem 2.2 χ(Gn,p ) = n/(r + O(1)) a.s. The key step in the proof is to establish ‘concentration of measure’. We find that a.s. the random graph Gn,p has the property that every induced subgraph with at least n/(log n)2 vertices contains a stable set of nearly r vertices; and so we can repeatedly remove such a stable set and colour it. The last few vertices can each get a new colour. The simple greedy or sequential colouring algorithmusually uses about twice as many colours as necessary. It is not known whether any polynomial-time algorithm usually achieves a ratio strictly less than 2. The situation is quite different for another model of random graph. Consider a fixed k, and suppose that the random graph Gn is obtained by sampling uniformly at random from the graphs on vertices v1 , . . . , vn which have no subgraphKk+1 . Then Gn is a.s. k-colourable, and there is a polynomial expected time algorithm which colours Gn optimally – see Prömel and Steger [30] and the references therein. 2 Edge colouring Vizing’s theorem says that the edge-chromatic number χ0 of a graph with maximum degree ∆ is equal to ∆ (Class 1) or ∆ + 1 (Class 2). It is of great interest (and NP-hard) to determine which case occurs for a given graph or class of graphs. How common are Class 2 graphs? Almost all graphs Gn,p have a unique vertex of maximum degree, and so are in Class 1. Thus Pr(χ0 = ∆ + 1) = o(1). In fact, this probability decreases to 0 very quickly [11]. Theorem 2.3 As n → ∞, 1 1 n−( 2 +o(1))n ≤ Pr(χ0 = ∆ + 1) ≤ n−( 8 +o(1))n . The lower bound here arises from regular graphs of odd order. The upper bound arises from a polynomial-time algorithm that attempts to edge-colour the graph with ∆ colours, and only rarely fails on Gn,p . It works by repeatedly removing near-perfect matchings. Very recently, Perkovic and Reed [28] have proposed an algorithm that will edge-colour Gn,p optimally in polynomial expected time. Total colouring For the total chromatic number χ00 , we can follow an approach similar to the above. It is clear that χ00 ≥ ∆ + 1. The well-known total colouring conjecture asserts that χ00 = ∆ + 1 or ∆ + 2. Theorem 2.4 As n → ∞, 1 Pr(χ00 > ∆ + 1) ≤ n−( 8 +o(1))n ; and, for some constant c with 0 < c < 1, 2 Pr(χ00 > ∆ + 2) = o(cn ). In the second inequality above, the bound on the probability is very small, and so we obtain some support for the total colouring conjecture, but we would prefer the bound to be zero! Connectedness and Hamiltonian cycles How many edges do we need to make a random graph be connected or be Hamiltonian? The threshold for connectivity in Gn,m was given by Erdős and Rényi [10] in 1960 in a seminal paper on random graphs. The proof involves estimating expected numbers of small components. Theorem 2.5 If m = 12 n(log n + cn ), and cn → c as n → ∞, then Pr(Gn,m is connected) → e−e 3 −c as n → ∞. It follows that if cn → −∞ then the probability → 0, and if cn → ∞ then the probability → 1. Determining the threshold for the existence of a Hamiltonian cycle was a major open problem for many years. The key idea turned out to be rotation-extension. Suppose that we have a path of length k, with vertices v0 , v1 , . . . , vk . If v0 or vk has a neighbour not in the path, then we can extend the path by simply adding such a neighbour. Failing this, suppose that vk has a neighbour vi , where 0 ≤ i ≤ k − 2. If i = 0 and there is an edge vi w joining the cycle v0 , v1 , . . . , vk , v0 to the rest of the graph, then the path w, vi , vi+1 , . . . , vk , v0 , . . . , vk−1 has length k + 1. If i 6= 0, then we perform a rotation and construct the path v0 , v1 , . . . , vi , vk , vk−1 , . . . , vi+1 ; this path still has length k, but has a different endpoint vi+1 from which we can again look for an extension. Using rotations and extensions Komlós and Szemerédi eventually solved the threshold problem in 1983 – see for example [2]. Theorem 2.6 If m = 12 n(log n + log log n + cn ), and cn → c as n → ∞, then Pr(Gn,m is Hamiltonian) → e−e −c as n → ∞. There is in fact a polynomial-time algorithm, based on the rotation-extension idea, that finds a Hamiltonian cycle with probability as in the theorem above – see Bollobás, Fenner and Frieze [5]. It has recently been shown that, for any fixed r ≥ 3, a random r-regular graph on n vertices is a.s. Hamiltonian – see Robinson and Wormald [31, 32]. The proofs are quite different, and involve a delicate use of the second moment method. The probabilities in the last two theorems are in fact the limiting probabilities that the minimum degree δ satisfies δ ≥ 1 and δ ≥ 2 respectively. Indeed there is a beautifully precise result, as follows. Consider the random graph process that starts with n isolated vertices and adds ³ ´ n the 2 possible edges one at a time, uniformly at random. The hitting time τn (δ ≥ 1) is the random number of edges that have to be added to make each degree at least 1, and similarly for other graph properties, such as being connected or Hamiltonian. The next result shows that, in the random graph process, the edge whose addition kills the last isolated vertex usually makes the graph connected, and similarly the edge whose addition brings the last vertex up to degree 2 usually makes the graph Hamiltonian. Theorem 2.7 τn (connected) = τn (δ ≥ 1) and τn (Hamiltonian) = τn (δ ≥ 2) a.s. 4 Minimum spanning trees Let G be any connected graph. Suppose that, for each edge e of G, we have a random length (or weight) Xe uniformly distributed on [0, 1], and these random variables are independent. Let msp(G) denote the expected value of the minimum total length of a spanning tree. The asymptotic behaviour of this quantity for the complete graph Kn was determined by Frieze in 1985, using early results of Erdős and Rényi on the emergence of a ‘giant component’ in sparse random graphs. Theorem 2.8 As n → ∞, msp(Kn ) → ζ(3), where ζ(3) = P∞ k=1 k −3 ∼ 1.2. For corresponding results for msp(Kn,n ), see [12]. Almost all Berge graphs are perfect A graph is perfect if for each induced subgraph the chromatic number equals the maximum clique size. A Berge graph is a graph with no induced subgraph isomorphic to an odd cycle or the complement of an odd cycle. All perfect graphs are Berge graphs: the strong perfect graph conjecture asserts that all Berge graphs are perfect. Let Pn be the number of perfect graphs, and let Bn be the number of Berge graphs, on vertices v1 , . . . , vn . Then Pn ≤ Bn and the strong perfect graph conjecture asserts that Pn = Bn . Prömel and Steger [29] have recently shown that Pn /Bn → 1 as n → ∞. 3 Random graphs: deterministic results We now consider some examples of the use of random graphs to obtain deterministic results. Ramsey numbers The Ramsey number R(k, l) is the least n such that, in any red-blue colouring of the edges of Kn , there is either a red Kk or a blue Kl . An easy induction shows that these numbers are finite, and indeed that R(k, k) ≤ (4 + o(1))k . What about lower bounds? Consider a random 2-colouring of the edges of Kn . The probability that there ³is´a monochromatic Kk is at most the expected number of such k k subgraphs, which is 2 nk 2−(2) . But this quantity is less than 1 if n ≤ 2 2 , for k ≥ 3. It follows that in this case there must exist a 2-colouring with no monochromatic Kk ; and so R(k, k) > 2k/2 . Indeed, the argument yields that R(k, k) = Ω(k2k/2 ) – see also Section 4 below. This beautifully simple proof is from Erdős in 1947 (see for example [1]). The lower bound has never been improved, apart from the constant factor (see the next section). 5 Graphs with large chromatic number and large girth The proof of the following theorem is another classic early example of the probabilistic method, due to Erdős in 1959 (see for example [1]). Deterministic constructions had to await Lovász in 1968. Theorem 3.1 For each k and g, there is a graph H with χ(H) ≥ k and girth(H) ≥ g. To prove the theorem it suffices to show that there is a graph G on n vertices such that √ (i) each set of s√= dn/ke vertices spans at least n edges, and (ii) fewer than n cycles have length less than g. For in this case we may obtain H by deleting an edge from each cycle in G of length less than g: clearly then H has girth at least g, and α(H) < s, so that χ(H) ≥ n/α(H) ≥ k. Is there a graph G as required? Consider Gn,p where p = 2k 2 /n, and n is large. The probability √ that property (i) fails is at most the expected number of s-sets spanning fewer than n edges, and this quantity is o(1). For property (ii), observe that, if En denotes the expected number of cycles of length less than g, then En ≤ g−1 X ni pi ≤ (2k 2 )g . i=3 √ Thus the probability that (ii) fails is at most En / n = o(1). Hence, for n sufficiently large, there is a graph G as required. The Hajós conjecture A natural strengthening of Hadwiger’s conjecture was the conjecture of Hajós in 1961, that if χ(G) = k then G has a subgraph homeomorphic to Kk . In 1979 Catlin produced counter-examples, but the conjecture was blown out of the water shortly afterwards by Erdős and Fajtlowicz (see [2]). Let p = 1/2. We have already seen that χ(Gn,p ) ≥ n/(2 log2 n) a.s. Let k = dn2/3 e . The probability that Gn,p has a subgraph homeomorphic to Kk is at most the probability that there is a k-set of vertices ‘missing’ at most n − k edges: but this is at most the expected number of such k-sets, which is o(1). Thus a.s. Gn,p has no subgraph homeomorphic to Kk . Hence, for all large n, there are graphs with chromatic number at least n/(2 log2 n) and no subgraph homeomorphic to Kk for any k ≥ n2/3 (which is much smaller). Expanders Expanders are very important in theoretical computer science – for example, in sorting networks and for economising in space or randomness. Roughly speaking, an expander is a graph in which each set S of vertices that is not too big has many neighbours outside S. There are many variations on the precise conditions. Generally it is easy to show that a suitable random graph is an expander with high probability, but hard to give an explicit construction (see for example [1, 2]). 6 4 The Lovász local lemma This section continues in the direction of the last one. The Lovász local lemma [9] is remarkably useful for proving existence results in graph theory and elsewhere. Let A1 , . . . , An be events in a probability space, and let G be a graph on the vertex set V = {1, . . . , n}. We call G a dependency graph for these events if, for each i ∈ V , the event Ai is independent of the family of all the other events Aj for j not adjacent to i. Theorem 4.1 (Lovász Local Lemma: symmetric version) Let A1 , . . . , An be events in a probability space Ω, with a dependency graph G. Suppose that each event Ai has probability at most p and that each vertex degree in G is at most d. If ep(d + 1) ≤ 1 T then P ( ni=1 Ai ) > 0. We discuss two applications of this result below: see [1] for extensions, references not given here and further applications. Hypergraph colouring The classic application [9] of the above result is to hypergraph 2-colouring. Theorem 4.2 Suppose that we have a family of subsets Ei of a set S, such that each subset Ei has size at least k and meets at most d other subsets Ej . If e(d + 1) ≤ 2k−1 then there is a 2-colouring of S such that none of the subsets Ei is monochromatic. This result gives a slight improvement in the lower bound for the Ramsey number R(k, k) discussed earlier: we take S as the edge set of Kn , and the subsets Ei of ³ ´ n S are the edge sets of the k possible subgraphs Kk . We obtain only a factor 2 improvement, but it gives the best bound known. To prove theorem 4.2, we colour the points of S blue or red independently and with equal probability. For each subset Ei , let Ai be the event that Ei is monochromatic. Then Pr(Ai ) ≤ 2−(k−1) . We may form a natural dependency graph G by letting distinct vertices i and j be adjacent just when the corresponding subsets Ei and Ej intersect. Then each vertex in G has degree at most d. But e(d + 1) 2−(k−1) ≤ 1, and so by the local lemma there must be a colouring as desired. The condition in theorem 4.2 can be improved slightly to e(d + 2) ≤ 2k (see [23]). It follows that, given a 7-uniform hypergraph (where each subset Ei has size 7 and each point is in 7 subsets Ei ), there must be a 2-colouring with no monochromatic edges. Recently Beck caused excitement when he gave an efficient general method to find a colouring as in theorem 4.2, although under more restrictive conditions on k and d. 7 Linear arboricity A linear forest is a forest in which each component is a path. The linear arboricity la(G) of a graph G is the minimum number of linear forests needed to cover its edges. The linear arboricity conjecture asserts that, if G has maximum degree d, then la(G) ≤ d(d + 1)/2e . It is clear that la(G) ≥ d/2, and for regular graphs it is easy to check that la(G) ≥ d(d + 1)/2e . The best asymptotic upper bound known for la(G) is due to Alon (see [1]). Theorem 4.3 There is a constant c such that, for every graph G with maximum degree d, 2 1 la(G) ≤ d/2 + c d 3 (log d) 3 . The proof uses the local lemma in different places. The following result supplies a key step. Note that any graph with n vertices and maximum degree d has a stable set of size at least n/(d + 1). At the cost of decreasing the size of the stable set by a constant factor, we can insist that it has a certain structure. Theorem 4.4 Let H be a graph with maximum degree d, and let V1 , V2 , . . . be pairwise disjoint subsets of the vertex set V , each of size s ≥ 2ed. Then there is a stable set W that contains a vertex from each set Vi . To prove this theorem, suppose that from each Vi independently we pick exactly one vertex uniformly at random, and let W be the random set of vertices picked. For each edge f of G, let Af be the event that W contains both end-vertices of f . Then each event Af has probability s−2 , and there is a natural dependency graph G in which each degree is less than 2sd. (There is a vertex of G for each edge of H, and the vertices of G corresponding to distinct edges f and f 0 of H are adjacent if some Vi contains an end-vertex of both these edges.) But es−2 2sd = 2ed/s ≤ 1. So, by the local lemma, there is some set W such that none of the events Af holds – that is, W is stable. By using Alon’s methods and the local lemma again, we can obtain a result on random graphs that lends some support to the linear arboricity conjecture [25, 27]. Theorem 4.5 (i) For fixed degree r, a random r-regular graph on n vertices has linear arboricity d(r + 1)/2e a.s. (ii) For fixed edge probability p, the random graph Gn,p has linear arboricity d∆/2e a.s. 5 Deterministic graphs: random methods Now we suppose that we are given some (deterministic) graph, and consider some random things we can do with it! 8 Finding a minimum-cut Let G be a connected graph with n (≥ 3) vertices. There are well-known efficient deterministic methods to find a minimum cut based on network flow ideas, but here is a far simpler randomised method [15, 17]. while at least three vertices remain choose an edge uniformly at random and contract it (keeping any multiple edges formed, but not loops). output the set of edges remaining. We claim that the probability that we obtain a minimum cut is at least 2/n2 , and so the probability that a minimum cut is not found in tn2 independent attempts is at 2 most (1 − 2/n2 )tn ≤ e−2t . To see why this claim is true, consider a fixed minimum cut C, of size k. The algorithm returns C if none of the edges in C is chosen. Now G has least nk/2 edges, since each vertex degree is at least k, and so the probability that the first edge chosen is not in C is at least 1 − (2/n). Conditional on this happening, the probability that the second edge chosen is not in C is at least 1 − 2/(n − 1) and so on. Hence the probability that C is returned is at least n−2 Y i=1 (1 − 2 2 )= . n−i+1 n(n − 1) Finding a maximum cut Suppose that we have a graph G = (V, E) with non-negative weights wij = wji on its edges. The maximum (weighted) cut problem is to find a set S of vertices that maximises the weight of the edges between S and V \ S. This problem is NP-hard. The following elegant random rounding idea has recently been shown to give good approximations [13]. The maximum cut problem may be modelled by the quadratic integer program max 12 X wij (1 − yi yj ) subject to yi ∈ {−1, 1} for all i ∈ V, i<j where we take G as Kn without loss of generality. Here yi = 1 corresponds to vertex i being in S say. We relax this program by replacing the variables yi ∈ {−1, 1} by variables vi ∈ Sn , where Sn is the unit Euclidean ball in <n . This gives the semidefinite program max 21 X wij (1 − vi .vj ) subject to vi ∈ Sn for all i ∈ V, i<j which can be solved in polynomial time. 9 The randomised method takes a (near) optimal solution v1 , . . . , vn to the relaxed program, picks a random vector r ∈ Sn , and outputs S = {i : vi .r ≥ 0}. It turns out that the expected weight of the corresponding cut is at least 0.878 times the optimal value. Similar ideas lead to an algorithm that will properly colour a 3-colourable graph using O(n1/4 ) colours [16]. Both of these algorithms are naturally randomised, but may be de-randomised to run in polynomial time and beat the best previously known such approximation algorithms. Random walks on graphs A random walk on a graph G starts at some vertex v0 , picks a neighbour v1 uniformly at random, from v1 picks a neighbour v2 uniformly at random, and so on. There are intimate connections between random walks and electrical networks. Given vertices u and v in G, let the commute time Cuv be the expected number of steps starting from u to reach v plus the expected number of steps starting from v to reach u; and let the effective resistance Ruv be the overall electrical resistance between u and v when each edge has resistance 1. Seemingly unrelated quantities like commute times and effective resistances satisfy the same harmonic relations, and we find that, if G is connected and has n vertices and m edges, then Cu,v = 2mRu,v : it follows that Cu,v < n3 [6]. Now suppose that we wish to test whether u and v are in the same component of G (when n is very large). Starting at vertex u, perform a random walk for 2n3 steps. If we ever reach v we say ‘yes’ (correctly) and if not we say ‘no’. By Markov’s inequality, the probability that we say ‘no’ incorrectly is less than 21 . This leads to a randomised algorithm that works in logarithmic space, since we need remember only where we are and how many steps we have taken. Random walks on (large) expander graphs G are particularly important. The random walk is ‘rapidly mixing’, in that it converges exponentially fast to the stationary distribution. If G is also regular then the stationary distribution is uniform. But then we can sample almost uniformly from the vertices of G, and from this we can approximately count the number of vertices. This approach allows us to estimate, for example, the number of perfect matchings in a dense graph [14], and to estimate the volume of a convex body[8]. The triangle problem It is NP-hard to determine whether there is a 2-colouring of the vertices of a graph G such that no triangle is monochromatic. But suppose that G has a proper vertex 3-colouring. Then clearly there must exist a 2-colouring of the vertices as above –just amalgamate two of the colours in a proper 3-colouring: the triangle problem for G is to find such a 2-colouring. Consider the following randomised ‘bit-flipping’ method [22]. 10 start with an arbitrary 2-colouring. while we find a monochromatic triangle pick a random vertex in the triangle, and flip its colour. What could be simpler? It is also effective: if G has n vertices, then we find a 2colouring with no monochromatic triangles in expected number of iterations at most 3 2 n . To see this, fix a proper vertex colouring of G with the colours -1,0 and 1, and 8 use the colours -1 and 1 for the 2-colouring: then half the dot product of the two colourings traces a symmetric random walk bounded within an interval of length n. There are good deterministic ways to solve the triangle problem using linear algebra, but there is no known ‘graph’ method [24]. Total colouring Let us return briefly to total colourings, and prove an upper bound on χ00 (G) by a simple randomised method [20, 26]. Theorem 5.1 Let G be graph with n vertices, and let k be an integer with k! ≥ n. Then χ00 (G) ≤ ∆ + k + 2. The algorithm starts with a vertex colouring with ∆ + 1 colours and an edge colouring with ∆ + 1 different colours. We pick a random bijection π between the two sets of colours, identify the corresponding colours, and let the ‘rejection graph’ R contain those edges that would give a colour clash in G. Now χ00 (G) ≤ ∆ + 1 + χ0 (R). But simple manipulations show that for each vertex, the probability that its degree in R is at least k + 1 is less than 1/k!. Hence Pr(∆(R) ≥ k + 1) < n/k! ≤ 1, and so for some bijection π we have χ0 (R) ≤ ∆(R) + 1 ≤ k + 1. The bound can in fact be tightened to χ00 (G) ≤ χ0 (G) + k [7]. To block or to direct? Two town planners, Blocker and Director, are bored and play a game on a graph G. Blocker flips a coin independently for each edge e and blocks (deletes) it with probability 12 . Let PB (hc) and PB (s → t) denote the corresponding probabilities that there remains an (unblocked) Hamiltonian cycle or path from s to t. Director similarly flips a coin for each edge e, and orients it so that each orientation is equally likely. Let PD (hc) and PD (s → t) denote the corresponding probabilities that there is a (directed) Hamiltonian cycle or path from s to t. How do these probabilities compare? It turns out [18] that PB (hc) ≤ PD (hc) and PB (s → t) = PD (s → t). We may prove these results by an induction involving the edges, using the following idea. Consider a particular edge e = {u, v} of G, and fix any pattern of blocking or orienting on all the other edges. If there is a Hamiltonian cycle not using e (and 11 respecting any orientations), then what happens with e is irrelevant, so we assume that this does not happen. Similarly we assume that there is a Hamiltonian path from u to v that does not use e, or one from v to u (or both). But then the probability that Blocker permits a Hamiltonian cycle is 12 , and the probability that Director does is either 12 or 1. 6 Concentration of measure and isoperimetric inequalities for graphs In 1979 Maurey used a ‘concentration of measure’ inequality to prove an isoperimetric inequality for ‘permutation graphs’ (see below). That started a ball rolling which has had exciting effects in the study of normed spaces, random graphs, the mathematics of operational research and theoretical computer science (see [19], which contains references not given here). This ball is still on the move – see [33]. Let us start with a special case of the ‘independent bounded differences inequality’. Theorem 6.1 Let X1 , . . . , Xn be independent random variables, with Xk taking valQ ues in a set Ak for each k. Suppose that the measurable function f : k Ak → < satisfies |f (x) − f (x0 )| ≤ 1, whenever the vectors x and x0 differ only in one coordinate. Let Y be the random variable f (X1 , . . . , Xn ). Then, for any t > 0, both Pr(Y − EY ≥ t) and Pr(Y − EY ≤ −t) are at most exp (−2t2 /n) . This result can be proved by elementary manipulations with the moment generating function MY (s) = E(esY ) of Y . It extends the standard Chernoff bound for binomial P random variables: to see this take each set Ak = {0, 1} and let f (x1 , . . . , xn ) = k xk . In the random graph Gn,p , let Xk give the edges present in the set of possible edges {vi , vk } with i < k. Then we obtain the following result. Theorem 6.2 Suppose that the graph function f satisfies |f (G) − f (G0 )| ≤ 1, whenever G0 can be obtained from G by changing edges incident with a single vertex. Then the random variable Y = f (Gn,p ) satisfies Pr(|Y − EY | ≥ t) ≤ 2 exp(−2t2 /n) for any t > 0. If we take f as the chromatic number χ, we see that χ(Gn,p ) is concentrated around its mean. Consider a finite metric space (V, d). We shall be interested in the case when V is the vertex set of a graph and d measures distance in the graph. For A ⊆ V and t ≥ 0, the t-neighbourhood At of A is the set {v ∈ V : d(v, A) ≤ t}, where d(v, A) = min{d(v, x) : x ∈ A}. A ‘discrete isoperimetric inequality’ gives a lower bound on |At | depending on |A| and t. 12 We shall focus on the n-cube Qn , in which the vertices are the 2n points in {0, 1}n , and two vertices are adjacent if they differ in exactly one coordinate. We shall also discuss the ‘permutation graph’ associated with the symmetric group Sn , in which the vertices are the n! permutations of {1, . . . , n}, and two vertices g and h are adjacent if g −1 h is a transposition. Theorem 6.3 Let A be any set of half of the 2n vertices of the n-cube Qn , and let q t0 = n/2. Then, for any t ≥ t0 , ³ ´ |At |/2n ≥ 1 − exp −2(t − t0 )2 /n . We can deduce this isoperimetric inequality from Theorem 6.1 by taking each Xk uniformly distributed on Ak = {0, 1} and letting f (x) = d(x, A) (see [19]). From a similar bounded differences inequality, we can obtain what is essentially the result of Maurey with which we started this section: we simply replace the phrase ‘the 2n vertices of the n-cube Qn ’ in Theorem 6.3 by ‘the n! vertices of the permutation graph of Sn ’. In the cube Qn , a Hamming ball centred at a vertex v and of radius d consists of all vertices w such that the distance from v to w is less than d, together with some vertices at distance d. A seminal theorem of Harper in 1966 (see [3]) concerns two sets A and C of vertices in Qn . The theorem asserts that there exist Hamming balls B0 centred at the all-zero vector and B1 centred at the all-one vector, such that we have |B0 | = |A|, |B1 | = |C| and d(B0 , B1 ) ≥ d(A, C). This yields the precise solution to the isoperimetric problem for the n-cube. From Harper’s theorem and a Chernoff bound, we can obtain the following cleaner version of Theorem 6.3. Theorem 6.4 Let A be any set of 2n−1 vertices of the n-cube Qn . Then, for any t ≥ 0, |At |/2n ≥ 1 − exp(−2t2 /n). The isoperimetric inequality Theorem 6.3 was obtained from Theorem 6.1 on concentration of measure. It is interesting that we can also reverse this process. Recall that a number L is a median of a random variable Y if both Pr(Y ≤ L) ≥ 21 and Pr(Y ≥ L) ≥ 21 . Theorem 6.5 Let X be uniformly distributed over V = {0, 1}n , let the function f on V satisfy |f (x) − f (x0 )| ≤ 1 whenever x and x0 differ only in one co-ordinate, and let L be a median of Y = f (X). Then, for any t ≥ 0, both Pr(Y −L > t) and Pr(Y −L < −t) are at most exp(−2t2 /n). This result is similar to Theorem 6.1 with ‘mean’ replaced by ‘median’. To see how it follows from Theorem 6.4, let A = {x ∈ V : f (x) ≤ L}. Then Pr(Y − L ≤ t) ≥ Pr(X ∈ At ) ≥ 1 − exp(−2t2 /n). 13 References [1] N. Alon and J.H. 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