The minimum size of a maximal strong matching in a random graph

Let be the random graph with fixed p 1. A strong matching S in Gn,p is a set of {el, e2, ... I em} such that no other of the connects an end-vertex of with an end-vertex of ej, ei ej. We show in this paper, that there exist positive constants Cl and C2 such that, with probability tending to 1 as n --+ 00, the minimum size of a maximal strong in Gn,p lies between 1/21ogdn c1logd1ogdn and logdn + c21ogdlogdn where d 1/(1 p). Let denote the random graph on n vertices with edge probability p fixed, o < p 1. Throughout this paper, we set d 1/(1 p). By the expression: "almost always", we mean: with probability tending to 1 as n --+ 00. A strong matching of Gn,p is set {ell .. ,em} of vertex-disjoint edges such that no other edge of the graph connects an end-vertex of ei with an end-vertex of ej, ii-j. In [2] we proved that, almost always, the maximum size of a strong matching in Gn,p achieves only a finite number of values. More precisely, we established the following theorem. Theorem 1 There exist positive constants Cl and C2 depending only on p and not on n I such that: 1) Almost always, Gn,p contains a strong matching of size m for each m satisfying m :::; logdn ~ logd1ogdn Cl' 2) Almost always, Gn,p does not contain a strong matching of size m for each m satisfying m ~ logd n ~ logd logd n + C2. Australasian Journal of Combinatorics .11.( 1995), pp. 77-80 The purpose of this paper is to evaluate the minimum size of a maximal strong matching in Gn .p . We shall prove the following theorem. Theorem 2 There exist positive constants Cl) C2, and not on n, such that and C4 depending only on p 1) Almost always} Gn,p has a maximal strong matching of size m for each m satisfying 1/21ogdn + c31ogdlogdn m::; logdn ~ logd1ogdn Cl 2) Almost always} fying m < 1/21ogd n has no maximal c410gd logd n or m size m for each m satislogd logd n + C2' We shall make use of the following lemma the tail of the binomial distribution, which can be deduced from Chernoff bounds. Lemma 1 Let Sn,p denote the binomial random variable with parameters nand p. Then} for any E 0 sufficiently small} we have P(ISn,p pnl ~ Epn) 2 Proof of 2 Let Xm denote the number of maximal Clearly, we have E(Xm) = ( n ) ( 2m 2m 2, ... ) m contained in where 7T' is the probability that any fixed UJ.(:!ov"'.LUiJl)':, of size m is a maximal matching in Gn,p' Let S be a fixed strong matching of size m. We denote N(S) the set of vertices which are not adjacent to any vertex of S. one can easily verify that S is maximal if and only if N(S) is either empty or an set. Moreover, we observe that IN(S)I is a binomial random variable with parameters n 2m and (1 _ p)2m. 2.1 The case m < ~ logd n a logd logd n We need to prove here that, if m < ~ logdn alogd1ogdn, where a is a positive constant which will be specified later, then E(Xm) tends to 0 as n -t 00. In this case, the expectation of IN(S)I satisfies E(IN(S)I) (n 2m)(1 p)2m ~ (logdn?a 0(1). Let A and B denote respectively the events "N(S) is stable" and {(I E)(logdn)2a S; IN(S)I S; n}. Clearly, we have