Random Graphs (Janson/Random) || Matchings

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<ul><li><p>4 Matchings </p><p>Perfect matchings play an important role in graph theory. On the one hand, they find a broad spectrum of applications. On the other hand, they are the subject of elegant theorems. The two results which characterize their exis-tence - Hall's and Tutte's theorems - are truly beautiful pearls of the theory. No wonder that Erds and Rnyi, after settling the question of connectivity (Erds and Rnyi 1959, 1961), turned their attention to the problem of find-ing the thresholds for existence of perfect matchings in random graphs (Erds and Rnyi 1964, 1966, 1968). </p><p>The results they obtained reveal a special feature of random graphs one could call "the minimum degree phenomenon." Namely, it is frequently true that if a minimum degree condition is necessary for a property to hold, then a.a.s. the property holds in a random graph as soon as the condition is sat-isfied. This phenomenon is discussed to larger extent in Section 5.1. Here we just mention that as soon as the last isolated vertex disappears, the ran-dom graph becomes connected (Erds and Rnyi (1959) and, for the hitting version, Bollobs and Thomason (1985)). Moreover, as we will learn later in this chapter (Corollary 4.5 and Theorem 4.6), provided n is even, from that very moment the random graph also contains a perfect matching (Erds and Rnyi 1966, Bollobs and Thomason 1985). </p><p>In the first section of this chapter we present a new proof of the threshold theorem for perfect matchings, which is based on Hall's rather than Tutte's theorem. This approach was originally designed to solve the problem of perfect tree-matchings (Luczak and Rucinski 1991), a special instance of (7-factors, where one looks for a vertex-disjoint union of copies of a given graph G which covers all vertices of G(n,p). Ordinary matchings correspond to the case </p><p>81 </p><p>Random Graphs by Svante Janson, Tomasz Luczak and Andrzej Rucinski </p><p>Copyright 2000 John Wiley &amp; Sons, Inc. </p></li><li><p>82 MATCHINGS </p><p>G A'2. Section 4.2 collects results on G-factors and partial G-factors in random graphs. Finally, in Section 4.3 we present recent advances on two long-standing open problems: finding the threshold for triangle-factors in G(n,p) and for perfect matchings in random 3-uniform hypergraphs, the latter known as the Shamir problem. </p><p>4.1 PERFECT MATCHINGS </p><p>A matching in a graph can be identified with a set of disjoint edges. A perfect matching is one which covers every vertex of the graph. Sometimes a perfect matching is called a 1-factor, because it is, in fact, a 1-regular spanning subgraph. A necessary condition for the existence of a perfect matching in a graph is the absence of isolated vertices. It turned out that in a random graph this is a.a.s. sufficient. Because of the simplicity of Hall's condition, it is quite straightforward to prove this result for random bipartite graphs. </p><p>Random bipartite graphs </p><p>Recall that a random bipartite graph, denoted here by G(n,n,p), is the relia-bility network with the initial graph being the complete bipartite graph Kn,n with bipartition (Vi,V^), |\ | = (V l = n. In other words, G(n,n,p) is ob-tained from Kn&gt;n by independent removal of each edge with probability 1 - p. Assume that logn log log n &lt; np &lt; 21ogn and suppose that the random graph G(n,n,p) does not have a perfect matching. Then, by the Hall Theo-rem, there is a set S C V for some i = 1,2, which violates Hall's condition, that is, | 5 | &gt; |7V(5)|, where N(S) is the set of all vertices adjacent to at least one vertex from S. Let 5 be a minimal such set. Then (Exercise!), </p><p>(i) \S\ = \N(S)\ + 1, </p><p>(ii) |S|</p></li><li><p>PERFECT MATCHINGS 83 </p><p>Fig. 4.1 A cherry in a graph marked with bold lines. </p><p>number of choices to realize (iii), we obtain </p><p>'M)sC)C-l)G)*"p!*"2(l-',),""'M, </p></li><li><p>84 MATCHINGS </p><p>size s &gt; 2 violating Hall's condition for any t [(logn-loglogn)/n, 2 logn/n], and the result follows easily (Exercise!). Remark 4.3. For future reference we explicitly state the error probability estimate </p><p>P(G(n,n,p) has no perfect matching) = 0 (ne - n p ) , (4.3) which is valid for all n and p. This too follows by the argument above; the probabilities of having an isolated vertex or a cherry are easily seen to be of this order (Exercise!), while a minor modification of (4.2) shows that, assuming, as we may, np &gt; log n, </p><p>PM) &lt; J ] e 2 , n ! , - 1 p 2 " ! e - r a p / ! = 0(n5p4e"3np /2) = 0(ne-" p ) . *&gt;3 </p><p>Ordinary random graphs </p><p>Let us return now to the ordinary random graph G(n,p), n even. Fixing an n/2 by n/2 bipartition of the vertex set and ignoring the edges within each of the two sets, we immediately see that Theorem 4.1 implies that G{n,p) has a perfect matching a.a.s. as soon as np 2logn oo. We will show how to reduce the above value of p by half so that, again, the threshold is the same as for the disappearance of isolated vertices (cf. Corollary 3.31). </p><p>In fact, we will show an even stronger result due to Bollobs and Thomason (1985). But first let us extend the notion of a perfect matching by saying that a graph satisfies property PM if there is a matching covering all but at most one of the nonisolated vertices. It can be routinely checked (Exercise!) that as soon as 2np- logn - log logn -* oo, there are only isolated vertices outside the giant component (cf. Chapter 5). Note that this holds already when the number of edges in G(n,p) is only about n logn - roughly half the threshold for the disappearance of isolated vertices. </p><p>However, the main obstacle for PM is now the presence of cherries. Two or more of them make it impossible. If there is exactly one cherry, PM is still possible, provided the number of nonisolated vertices (as well as isolated vertices) is odd. The expected number of cherries is </p><p>3 M p 2 ( 1 _ p )2(n-3) &lt; n 3 p 2 e -2n P + 6p = o ( 1 ) </p><p>if 2np - logn - 2 log logn -&gt; oo, which also holds already when there are about nlogn edges in G(n,p). Again, as proved by Bollobs and Thomason (1985), this trivial necessary condition becomes a.a.s. sufficient. Theorem 4.4. Let yn = 2np - logn - 2 log logn. 77ien </p><p>0 if yn -+ -oo, </p><p>(l + ^e - c ) e -* ' " c ifyn^c, 1 t /yn-&gt;oo. </p></li><li><p>PERFECT MATCHINGS 85 </p><p> e- oo, if np log n c, j / n p log 7i &gt; oo. </p><p>Consequently we obtain a result of Erds and Rnyi (1966). </p><p>Corollary 4.5. </p><p>P(G(n,p) has a perfect matching) </p><p>The proof below is easily modified to give the corresponding hitting time result too (Exercise!). </p><p>Theorem 4.6. The random graph process {G(n, M)}M is a.a.s. such that the hitting times = min{M : (G(n, M)) &gt; 1} and rp m = min{M : G(n, M) has a perfect matching} coincide. </p><p>The original proofs of the 1-statements of Theorem 4.4 and Corollary 4.5 were both based on Tutte's theorem. Here we propose an alternative approach, via Hall's theorem, by Luczak and Rucinski (1991). This approach relies on the following technical lemma. Given two disjoint sets of vertices in a graph, the bipartite subgraph induced by them consists of all edges with one endpoint in each set. </p><p>Lemma 4.7. Let np = 0( logn) . For every c &gt; 0, a.a.s. every bipartite subgraph, induced in G(n,p) by two sets of equal size and with minimum degree at least clogn, contains a perfect matching. </p><p>Proof. Set u = "''"L"* By t n e first moment method it is easy to check (Exercise!) that a.a.s. </p><p>(i) for every pair of disjoint subsets of vertices of size bigger than u there is in G(n,p) an edge between them, </p><p>(ii) every set S of at most 2u vertices induces in G(n,p) fewer than (loglogn)3 |S| edges. </p><p>The rest of the proof of Lemma 4.7 is purely deterministic. Suppose G is an n-vertex graph satisfying (i) and (ii), and B is a bipartite subgraph of G induced by the bipartition (Wi,W2), |Wi| = \W2\ = w, () &gt; clogn, but without a perfect matching. Then, by Hall's theorem there is 5 C W\ such that \NB(S)\ &lt; \S\. Case 1: \S\ &lt; u. Then \S U N B ( 5 ) | &lt; 2u, but since all the edges with one endpoint in S have the other endpoint in NB(S), there are at least c logn |5 | such edges a contradiction with (ii). Case 2: \NB(S)\ &gt;w-u. Then \Wi \S\ &lt; \W2 \ NB(S)\ &lt; u and all the edges with one endpoint in W2 \NB(S) have the other endpoint in Wi \S - again a contradiction with (ii). </p></li><li><p>86 MATCHINGS </p><p>Case 3: \S\ &gt; u,\NB(S)\ u, but there is no edge between S and W2 \ NB(S) - a contradiction with (i). </p><p>Proof of Theorem 4-4- Throughout the proof we assume that | logn &lt; np &lt; 21ogn and n is even. </p><p>If J/n -* oo, the second moment method yields that a.a.s. there are many cherries in G(n,p) (Exercise!). Since already the presence of two cherries makes PM impossible, the O-statement follows. </p><p>If Vn -* c, the number of cherries has asymptotically the Poisson distribu-tion with expectation ge~c. It can also be proved that the probability that there is exactly one cherry and the number of isolated vertices is odd, con-verges to ^ e - c e x p ( i e - c ) . (Advanced Exercise! Hint: Apply the two-round exposure - c/. Section 1.1 - with p? so small that during the second round a fixed cherry cannot be destroyed.) </p><p>If Vn - oo, then, as was shown above, a.a.s. there are no cherries at all. Thus, in order to prove the two latter parts of the theorem, we have to show that the only obstruction for PM is the presence of either at least two cherries or one cherry while the number of isolated vertices is even. </p><p>The idea of the proof is as follows: Suppose that we have a graph on n vertices which has either no cherries at all or exactly one cherry, but then the number of isolated vertices is odd. Fix an arbitrary bipartition of the vertex set into two halves called sides. Call a vertex bad if it has either fewer than 2o log n neighbors within its own side or fewer than 355 log n neighbors on the other side. Assume that the graph satisfies the hypothesis of Lemma 4.7 (say, with c = 555) and some other properties held a.a.s. by G(n,p) {viz. Claim below). </p><p>Match the bad vertices first (except for the isolated vertices, of course). If there is an odd number of them, leave one out. If the graph has a cherry, the left-out vertex must be a degree one vertex of that cherry. Remove all bad vertices and their partners (which do not need to be bad) from the graph. Then adjust the bipartition so that it becomes even again, but so that the minimum degree of the induced bipartite subgraph has not dropped much. Finally, apply Lemma 4.7 and obtain a matching covering all but at most one of the nonisolated vertices. </p><p>Now we turn to the details. Let X be the number of bad vertices in G(n,p), and let Y Bi( ,p). Then E(X) = nP(vertex 1 is bad) and, by (2.6), for sufficiently large n, </p><p>P(vertex 1 is bad) &lt; 2P(F &lt; ^ l o g n ) &lt; n"0 , a i . </p><p>Hence, by Markov's inequality (1.3), a.a.s. X &lt; n4/5. We still need to distinguish another class of vertices of low degree. Call a </p><p>vertex small if its degree in G(n,p) is at most three. We claim that neither small nor bad vertices can cling together too much. </p></li><li><p>PERFECT MATCHINGS 87 </p><p>Claim. For every fixed integer k, a.a.s. there is no k-vertex tree in G(n,p) which contains more than two small vertices or more than four bad vertices. </p><p>Proof. Let Z e Bi(n - Jfc + l ,p). It is straightforward to show (Exercise!) that for every fixed z </p><p>?(Z </p></li><li><p>88 MATCHINGS </p><p>Uii 2 0 </p><p>(a) (b) </p><p>Fig. 4.2 Scenes from the proof of Theorem 4.4. </p><p>If 2 &lt; deg(u) &lt; 3, then v has at most one neighbor within the set Vj_i, since otherwise there would be three small vertices on a small tree (Exercise!). Thus we may choose u as one of the neighbors of Vi outside V_i. </p><p>If deg(rjj) &gt; 4, then Vi has at most three neighbors within the set V_i, since otherwise there would be five bad vertices on a small tree (Exercise! -see Figure 4.2(a)). Again, we may choose u as one of the neighbors of v outside V_!. </p><p>Hence, a.a.s. all nonisolated bad vertices can be matched. The removal of bad vertices and their partners does not affect the degrees of other vertices by more than eight (Exercise! - see Figure 4.2(b)). However, the remaining vertices may no longer form an even bipartition. In order to apply Lemma 4.7 we have to balance them back by moving across up to n4^5 carefully chosen vertices. To minimize the effect on degrees in the induced bipartite graph, we use for this purpose a 2-independent set of vertices, that is, an independent set of vertices no two of which have a common neighbor (thus, the degree of a vertex may drop further by at most one). Trivially, there is always such a set of size at least /(2 + 1) (Exercise!), which is more than is needed. The </p></li><li><p>G-FACTORS 89 </p><p>obtained bipartite subgraph has, therefore, minimum degree at least </p><p> b e n - 9 &gt; -r- logn, 200 300 and, by Lemma 4.7, it contains a perfect matching, which together with the previously constructed matching {v\,u\},..., {u/,Uf} forms a matching cov-ering all but at most one of the nonisolated vertices. This completes the proof of Theorem 4.4. </p><p>Disjoint 1-factors </p><p>Erdos and Rnyi, after establishing a threshold for the existence of at least one perfect matching in the bipartite random graph G(n, n,p), went on and gener-alized their result to the existence of at least r disjoint 1-factors in G(n,n,p) (Erds and Rnyi 1966). Trivially, if a graph possesses fewer than r disjoint 1-factors, then the removal of all the edges of a maximal family of disjoint 1-factors results in a graph with no 1-factor at all and with all the degrees decreased by at most r 1. Thus, if the original graph had minimum degree at least r, then, after the removal, there would be no isolated vertices left, and Hall's condition would have to be violated in a nontrivial way. By an argument similar to that used in the proof of Theorem 4.1 this can be shown to be unlikely for G(n,n,p) (Exercise!), and it follows that the threshold for containing at least r disjoint 1-factors in G(n,n,p) coincides with that for minimum degree r. The latter can easily be found by the method of moments (Exercise!). The respective hitting time result holds too. </p><p>The corresponding problem for an ordinary random graph G(n,p) was solved much later by Shamir and Upfal (1981) via an algorithmic approach. Since every even Hamilton cycle is a union of two disjoint 1-factors, the so-lution also follows, together with the hitting time version, from a result of Bollobas and Frieze (1985) (see Section 5.1). However, the proof of Theo-rem 4.4 presented above can easily be adapted to yield the threshold for disjoint 1-factors as well. It boils down to showing that a.a.s. after removing the edges of an arbitrary subgraph of maximum degree at most r 1, the remainder of G(n,p) will contain a perfect matching. The definition of a bad vertex remains unchanged, while a small vertex is now one with degree at most r + 2. The details are left to the reader (Exercise!). It follows that a.a.s. the hitting time for having r disjoint 1-factors coincides with the hitting time for having minimum degree at least r. </p><p>4.2 G-FACTORS </p><p>In this section we study thresholds for containment of spanning (or almost spanning) subgraphs in G(n,p) which are unions of vertex disjoint copies of a given graph. For a gr...</p></li></ul>