In this section we will show that when the number of edges in Gmin(n, M) is larger than the number of vertices, a.a.s. there is only one largest component with more than a half of the vertices, and all other possible components are small, i.e., of at most logarithmic order, and are unicyclic.
Theorem 12.3.1. Letδ >0. Then with probability1−O(1/n)each component of Gmin(n,(1 +δ)n)smaller thann/2 has at most(2/δ) lognvertices and con-tains at most one cycle. Moreover, for every functionω=ω(n)→ ∞, a.a.s. the number of all vertices contained in unicyclic components ofGmin(n,(1 +δ)n)is smaller thanω.
Proof. We first note that the probability that, for somek, (2/δ) logn≤k≤n/2, a graphGmin(n,(1+δ)n) contains a component withkvertices (and hence with at leastk−1 edges) is bounded from above by
n/2
where the first factor counts all possible choices of the vertex setS of a compo-nent ofk vertices, the second one bounds the probability that every edge with one end inS has the other end inS too, and the last factor is the probability that every vertex outsideS has chosen the other end outsideS. Note also, that
n
12.4. CONNECTEDNESS 163
Hence, for large enoughn, one can bound (12.3.1) from above by
n/2
Note that any component with at least two cycles has more edges than vertices. Hence the probability that a component of Gmin(n,(1 +δ)n) smaller thand(2/δ) lognecontains at least two cycles is, for n large enough, bounded by
Hence, from (12.3.2) and (12.3.3) it follows that the probability that a com-ponent of Gmin(n,(1 +δ)n) smaller than n/2 contains at least two cycles is O(1/n).
Finally, let ω =ω(n)→ ∞, and letUk be the number of unicyclic compo-nents of k vertices inGmin(n,(1 +δ)n). Note that a unicyclic component has the same number of vertices and edges and thus
n
Thus, the probability that at leastωvertices ofG(n,(1+δ)n) belong to unicyclic components is, by Markov’s inequality, bounded above byO(1/ω) =o(1).
Theorems 12.2.1 and 12.3.1 imply that the giant component must have ap-peared when the number of edges is betweenh1nand nwithh1 .
= 0.6931. We will show in Section 12.7 that it happens, in fact, when the number of edges becomeshcrnwithhcr
= 0. .8607.
12.4 Connectedness
In this section we study how the probability that Gmin(n, M) is connected changes as M grows. The main result of this section determines this proba-bility quite precisely for most of the stages of the processGmin(n, M).
164 CHAPTER 12. MINIMUM DEGREE PROCESS
Theorem 12.4.1. Let constants h2,h3 be defined as in (12.1.1) and let ρn(t) denote the probability that Gmin(n, tn) is connected. Then, for every constant t6=h2, the limit
ρ(t) = lim
n→∞ρn(t)
exists and ρ(t) = 0 fort < h2 while ρ(t) = 1 fort ≥h3. If t ∈(h2, h3), then 0< ρ(t)<1, where
ρ+= lim
t→h2+ρ(t)>0 and lim
t→h3−ρ(t) = 1.
Proof of Theorem 12.4.1. For eacht < h2, Theorem 12.2.1 implies Gmin(n, tn) a.a.s. contains many isolated edges and so it is a.a.s. disconnected; henceρ(t) = 0 fort < h2.
Ift > h2then, by Theorems 12.1.1 and 12.3.1, a.a.s. Gmin(n, tn) consists of one large component and, perhaps, some short isolated cycles.
In particular, ift > h3, then Theorem 12.1.1 implies a.a.s.δ(Gmin(n, tn))≥3 and thusGmin(n, tn) contains no isolated cycles. Consequently, fort > h3a.a.s.
Gmin(n, tn) is connected andρ(t) = 1.
It is enough to consider the case t ∈(h2, h3]. Let us fixt ∈(h2, h3). Note that from Theorem 12.1.1 it follows that for some functionω=ω(n)→ ∞with probability 1−O(ω−2) for the process{Gmin(n, M)} the following holds:
(i) |H2−h2n| ≤n/ω3;
(ii) forM =H2, we have|X2(n, M)−α2(h2)n| ≤n/ω3; (iii) |X2(n, tn)−α2(t)n| ≤n/ω3.
In our further argument we shall often condition on the eventBthat (i)–(iii) hold for (Gmin(n, M))M≥0. Note that, since Pr(B) = 1−O(ω−2), for any event Awe have
Pr(A|B) = Pr(A)−O(ω−2)
1−O(ω−2) =O(ω−2) + (1−O(ω−2)) Pr(A). (12.4.1) Let Zk = Zk(n, tn), k = 2,3, . . ., denote the number of isolated cycles of length k in Gmin(n, tn). We first estimate the expectation of Zk in the conditional probability space, when we condition on B. In Gmin(n, tn) there exist nkcandidates for the set of vertices of an isolated cycle of lengthk. Let us fix one such subsetS. Note that ifGmin(n, tn) contains an isolated cycle with vertex setS, then all of its edges appear inGmin(n, M) already at the moment H2, when the minimum degree of a graph reaches two. If at this moment a cycle is isolated, then each time we chose the first end of an edge outsideS we had to pick as the second end of an edge a vertex outsideS as well. By (12.4.1) the probability of that event is given by
O(ω−2) + (1 +O(ω−2))n−k−1 n−1
(1+O(ω−3))h2n−k
=O(ω−2) + (1 +O(kω−3+k2/n))e−h2k.
(12.4.2)
If Gmin(n, M) contains an isolated cycle on vertex set S, it means that until this moment each time we have picked up one end of an edge in S the second
12.4. CONNECTEDNESS 165 end has been chosen also inS, in such a way that it created with edges which had already been selected a forest which consisted of paths, and, eventually, a cycle of lengthk. Thus, the probability that inG(n, H2) the subsetS spans a Hence the probability that, conditioned onB, there exists an isolated cycle on S inG(n, H2) is given by a product of (12.4.2) and (12.4.3).
The probability that a cycle on the set S which is isolated at the moment M =H2 remains isolated also inGmin(n, tn) is the probability that each vertex of the cycle has degree two also inGmin(n, tn). It is easy to see that if byW2and W2(t) we denote the sets of vertices of degree two inGmin(n, M) at the moments M =H2andM =tnrespectively, then each subset ofW2of|W2(t)|elements is equally likely to becomeW2(t) later in the process. Hence the probability that W2(t) contains a given subset ofS⊆W2ofk elements is equal to
Consequently, the probability that, conditioned on B, an isolated cycle on S present inGmin(n, M) for M =H2remains isolated also inGmin(n, tn) is given cycles longer thanω, a.a.s.Z(n, tn) is equal to the number of all isolated cycles ofGmin(n, tn). From (12.4.4) we infer that
166 CHAPTER 12. MINIMUM DEGREE PROCESS In particular, Theorem 12.3.1 implies that
Pr(Gmin(n, tn) is connected) = (1 +o(1)) Pr(Z(n, tn) = 0) nthe probability that Gmin(n, nt) is connected is a non-decreasing function of t, we haveρ(h3) = 1.
Observe that the limit behaviour of the minimum degree multigraph pro-cess (Gmin(n, M))M is very different from the classical random graph process (G(n, M))M mentioned in Section 3.1, in which a.a.s. G(n, M) becomes con-nected for M = n2(logn+ω(n)) with ω(n) → ∞. However, it is worthwhile to compare Theorem 12.4.1 with analogous results for two other random graph process models in which the minimum degree grows quickly with the number of edges.
The first one is the uniform graph process (U(n, M))M, in which theMth edge ofU(n, M) has one end at vertexM− bM/ncwhile its other end is chosen uniformly at random from alln−1 possibilities (the vertex set of U(n, M) is {0,1, . . . , n−1}). Jaworski and Łuczak [83] proved that for every t ≥ 0 the chosen uniformly at random among all pairs of vertices esuch that the graph Gd(n, M−1)∪ehas the maximum degree at mostd. It was shown by Ruciński and Wormald [124] that if 0 ≤ t < d/2 then a.a.s. Gd(n, tn) is disconnected.
12.5. TWO PHASES 167