On a correlation inequality of Farr

Transcription

1 On a correlation inequality of Farr Colin McDiarmid, Department of Statistics Oxford University Abstract Suppose that each vertex of a graph independently chooses a colour uniformly from the set {1,..., k}; and let S i be the random set of vertices coloured i. Farr shows that the probability that each set S i is stable (so that the colouring is proper) is at most the product of the k probabilities that the sets S i separately are stable. We give here a simple proof of an extension of this result. 1

2 1 Introduction Given a graph with vertex set V and given a positive integer k, consider a random k-colouring of V, where each vertex independently chooses a colour from the set {1,..., k}. For each i let S i be the random set of vertices coloured i. For the special case when colours are always chosen uniformly from {1,..., k}, Farr [3] gives the following correlation inequality: ( k ) P {S i stable} i=1 k P {S i stable}. i=1 Recall that a set of vertices is stable (or independent) if no two are adjacent, and so the left hand side above is the probability that the random colouring is proper. The above inequality immediately gives a bound on the chromatic polynomial of the graph in terms of the stability polynomial which is not quite so hard to calculate - see [3]. This inequality is attractive and intuitively plausible (though by no means obviously true). Farr proves it using the powerful Ahlswede-Daykin four functions inequality [1]. We shall give a short and simple proof of an extension, based on an inequality of Harris [5]. A family F of sets is decreasing (or hereditary) if any subset of a set in F is also in F. Thus the family of stable sets in a graph is decreasing. In a similar way we may define an increasing family of subsets of a set. 2

3 Theorem Let V and I be finite non-empty sets. Let (X v : v V ) be a family of independent random variables, each taking values in some set containing I; and for each i I let S i = {v V : X v = i}, so that S i is the random set of elements coloured i. Let (F i : i I) be a family of collections of subsets of V such that each is increasing or each is decreasing. Then ( ) P {S i F i } P {S i F i }. i I i I Suppose that the random variables X v are identically distributed. When each set F i is the collection of stable sets in a given graph we obtain Farr s result. When F i is the collection of subsets of V of size at most x i we see that the multinomial distribution is negatively orthant dependent. We shall deduce the theorem from a general lemma which asserts that certain events F and G are negatively correlated. 2 A general lemma Let T = (T i : i I) be a family of independent binary random variables. Let F be a decreasing collection of subsets of I, and let G be an increasing collection of subsets of I. Then the events F = {T F} and G = {T G} are negatively correlated, that is P (F G) P (F )P (G). (We are identifying a set and its incidence vector.) This is the elementary but very useful inequality of Harris [5], which is one of the basic inequalities in a family of correlation inequalities headed by the Ahlswede-Daykin four-functions inequality [1] -see for example [2, 4, 8]. Below we formulate a general lemma which asserts that certain events 3

4 are negatively correlated, and we observe that this lemma yields the theorem; and in the next section we prove the lemma, using the inequality of Harris. Let us introduce the lemma. Let the finite set I be partitioned into two non-empty sets J and K. Let F be a collection of families (W j : j J) of subsets of V which is decreasing in the natural componentwise sense, and similarly let G be a decreasing collection of families (W k : k K). As in the theorem, let (X v : v V ) be independent random variables, and for each i I let S i = {v V : X v = i} so that S i is the set of elements v coloured i. It should be intuitively plausible that the events F = {(S j : j J) F} and G = {(S k : k K) G} are negatively correlated, and the lemma states that this is indeed the case. Lemma P (F G) P (F )P (G). It is easy to check that if two events are negatively correlated then so are the complementary events. Before we go on to prove the lemma let us state an immediate corollary, which yields the theorem by an easy induction on the size of I. Corollary of I into non-empty sets J and K. Then With premises as in the theorem, consider a partition ( ) P {S i F i } P {S j F j } P {S k F k }. i I j J k K This last result (in the decreasing case) also follows easily from the general clutter percolation theorem [6, 7] - and indeed it was first proved that way - but the proof here is simpler and more direct. 4

5 3 Proof of lemma The idea of the proof is to consider generating the random colouring (X v : v V ) in two stages, by first deciding for each v V whether X v J or X v K. It is convenient to assume that each X v takes values in I. We may do this without loss of generality: for we could add a new element j 0 to J forming J + ; replace F by F + where (W j : j J + ) F + if and only if (W j : j J) F; and replace X v by X v +, where X v + = X v if X v I and X v = j 0 otherwise. For each v V, let the random variable T v have distribution given by P {T v = 1} = P {X v J} and P {T v = 0} = P {X v K} = 1 P {T v = 1}. Also, let Y v have distribution that of X v given X v J, and let Z v have distribution that of X v given X v K. (If P {X v J} = 0 let Y v = 0, and similarly for Z v.) Further assume that all these 3 V random variables are independent. Clearly we may assume that the random colouring (X v : v V ) is generated from T = (T v : v V ), Y = (Y v : v V ) and Z = (Z v : v V ), so that X v = i if i J, T v = 1, Y v = i or i K, T v = 0, Z v = i. For j J we may write the jth colour set S j as S j (T, Y) where S j (t, y) = {v V : t v = 1, y v = j}; and similarly for k K we may write S k = S k (T, Z) where S k (t, z) = {v V : t v = 0, z v = k}. 5

6 Consider a fixed y J V and z K V. Let F (y) = {t {0, 1} V : (S j (t, y) : j J) F}, and note that F (y) is decreasing. Similarly G(z) = {t {0, 1} V : (S k (t, z) : k K) G} is increasing. Hence by the Harris inequality and the independence of T, Y, Z we have P (F G Y = y, Z = z) = P ({T F (y)} {T G(z)}) P {T F (y)}p {T G(z)} = P (F Y = y)p (G Z = z). Finally then P (F G) = y z P (F G Y = y, Z = z)p (Y = y)p (Z = z) y z P (F Y = y)p (G Z = z)p (Y = y)p (Z = z) = P (F )P (G). 6

7 References [1] R.Ahlswede and D.E.Daykin, An inequality for the weights of two families of sets, their unions and intersections, Z. Wahrlichkeitstheorie verw. Gebeite 43 (1978), [2] B.Bollobás, Combinatorics, set systems, hypergraphs, families of vectors and combinatorial probability, Cambridge University Press, [3] G.E.Farr, A correlation inequality involving stable set and chromatic polynomials, J.Combinatorial Theory B, to appear. [4] R.L.Graham, Applications of the FKG inequality and its Relatives, Mathematical Programming The State of the Art, Springer-Verlag, 1983, [5] T.E.Harris, A lower bound for the critical probability in a certain percolation process, Proc.Camb.Phil.Soc.56 (1960), [6] C.J.H.McDiarmid, Clutter percolation and random graphs, Mathematical Programming Study 13 (1980), [7] C.J.H.McDiarmid, General percolation and random graphs, Advances in Applied Probability 13 (1981), [8] C.J.H.McDiarmid, Some inequalities for random graphs and percolation, Notes from New York Academy of Science Graph Theory Day 13 (1987),