Matchings in Random Spanning Subgraphs of Cubelike Graphs

Alexandr V. Kostochka Institute of Mathematics of the Siberian Branch of the Academy of Sciences of the USSR, 630090 Novosibirsk, USSR

ABSTRACT

A question about the evolution of random spanning subgraphs G, of bipartite regular so called cubelike graphs G is considered. It is shown that for G , of any large enough cubelike graph G the threshold to have a 1-factor is the same as the threshold to have no isolated vertices. This generalizes a conjecture of K. Weber.

1. INTRODUCTION

Let QZn be a random spanning subgraph of the n-cube Qn such that the edges of Q” belong to Q;n with probability p , independently of each other. It is known [4,5,9] that

lim P(Q; has no isolated vertices) = ”+@=

Weber [lo] conjectured that for any p >0.S

lim P(Q; contains a perfect matching) = 1 n-+m

It was proved in [3] and [6] independently that Weber’s conjecture is true. It will be shown in the present paper that this conjecture is true not only for the sequence {Q“},“=, but for many others. Besides, we will see that if p is large enough to provide the absence of “cherries” in the Q;, then almost surely has a matching covering all nonisolated “even” vertices or all nonisolated “odd” vertices. So, the situation is similar to the situation with Kn,p (see [2], p. 160).

Random Structures and Algorithms, Vol. 1, No. 3 (1990) 0 1990 John Wiley & Sons, Inc. CCC 1042-9832/90/030277-09$04.00

278 KOSTOCHKA

2. NOTATION

For any graph G we denote by V ( G ) , E(G), and A(G) the set of vertices, the set of edges, and the maximal degree of G, respectively. Usually, graph G is denoted by the pair (V, E), where V = V ( G ) and E = E(G). Sometimes a bipartite graph G with parts X and Y will be denoted by (X, Y; E) where E = E(G).

Given A C V ( G ) , let G ( A ) denote the induced subgraph of G on A and N,(A) denote the neighborhood of A , i.e., { u E V(G)\Al3u E A : (u, u ) E E ( G ) } . For A , B C V ( G ) we let

E,(A, B ) = { (u , u ) E E(G)(u E A , u E B }

The term Q" denotes the n-cube; the term logx means log, x. Let G be a graph, 0 ' p 5 1. Then G, is a random variable whose values are spanning subgraphs of G such that for any e E E(G) we have P(e E E(G,)) = p and the events e E E(G,) are totally independent for distinct edges.

3. CUBELIKE SEQUENCES

Let k, m be natural numbers. Define for n 2 k the family %"(k, m) of graphs by induction on n. The family V$(k, m) consists of all connected bipartite k-regular graphs on m vertices. Every G E %"(k, m) for n 2 k + 1 is constructed as follows. We take two graphs G', G" E %n-l(k, m) on disjoint vertex sets and add a 1-factor every edge of which connects V(G') with V(G") so that the obtained graph is bipartite.

Clearly, Q" E % " , ( l ~ , 2 ~ ) for any n z k . The graph H presented in Figure 1 belongs to %4(0, 1) fl %4(1, 2) Ti %4(2, 4) and is not isomorphic to Q4.

Lemma 0. graph; ( b ) IV(G)l= m2"-k; (c ) if 1 5 i 5 IV(G))/2, A C V ( G ) , ]A1 = i, then

Let G E %,,(k, m). Then (a ) G is an n-regular bipartite connected

IE,(A, V(G)\A)I 2 i /m .

Proof. The statements (a) and (b) are obvious. We prove (c) by induction on n for fixed k and m. Suppose (c) is true for all k 5 n < n o and consider G C %n,o(k, m) and A C V ( G ) with

0 < IA1= i 5 0.51V(G)\ . (1)

If no = k, then, due to connectivity of G, we have (E,(A, V(G)\A)I 2 1 > i /m. Now, let no > k. By the definition, V ( G ) = V, U V,, where V, f l V, = 0, G(V,) E

%no-l(k, m) for j = 1,2 and the set EG(Vl, V,) is a 1-factor in G. Let us denote A, = A f l V,, i, = lAjl for j = 1,Z. We may suppose i , 5 i , . Then by the choice of no

/,!?,(A,, V,\A,)( 2 i , /m , (,!?,(A,, V,\A,)( 2min{i,, IVll - i , } / m .

MATCHINGS IN RANDOM SPANNING SUBGRAPHS OF CUBELIKE GRAPHS 279

Fig. 1.

Besides, IE,(A,, V,\A,)I 1 i, - i, 2 (il - i2)/m. Hence

(E,(A, V(G)\A)( L i,lm + min{il, IVll - il}/m + (il - i,)/m

2 min{2i,, IV,/} /m .

Since 2i1 2 i and due to (l), i 5 / V l / ; the Lemma is proved. m

Definition. A multigraph G will be called a cubelike graph (of degree n) if

i. G is an n-regular multigraph; ii. every vertex of G is adjacent to at least n -log n vertices; iii. for any A C V ( G ) with n 5 ( A ( = i I (V(G)(/Z

(E,(A, V(G)\A)( 2 3 n + i(10g4 n)ln ;

iv. n-'2" 5 IV(G)lS n2". A sequence { G"},"=, is said to be a cubelike sequence if there is no such

that, for any n 2 no, the graph G" is a cubelike graph. A cubelike sequence { G"}:=, will be called a strongly cubelike sequence if there are numbers n1 and c such that for any n 2 n1

iv'. cP12" s IV(G")( cr c2".

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Examples.

1. By Lemma 0 for arbitrary m and k any sequence {G"},",, such that G" E %"(k, m) ( n = k , k + 1, - . . ) is a strongly cubelike sequence.

2. Let 1x1 = 1 YI = 2"-', X f l Y = 0. Drop at random n perfect matchings, every edge of which joins X with Y. By Konig's theorem, any n-regular bipartite multigraph with parts X and Y can be obtained as a result of this procedure. On the other hand, it is not hard to prove that almost surely we obtain a cubelike graph.

3. We can also use a sort of Bollobas' configurations (see [2], pp. 48-49) for modeling n-regular bipartite multigraphs on 2" vertices and once again almost surely cubelike graphs will be constructed.

4. RESULTS

The main result is the following.

Theorem 1. { p,},", , be a sequence of real numbers, O s p , I 1, such that

Let { G"},"=, = {(X", Y"; E")},",, be a cubelike sequence and

IV( G ")In2( 1 - pn)'" - 0 n-m

Denote by X:" (resp. Y"p,) the set of vertices of X" (resp. Y " ) which are nonisolated in G i n . Then almost surely Gin contains a matching with min{lX",l, lY",l} edges.

Theorem 1 implies immediately

Corollary 1. real numbers, 0 ~ p , , 5 1, such that

Let { G"}:=, be a cubelike sequence and { p,,}:=, be a sequence of

lim P(Gin has no isolated vertices) > 0 , "-30

Then

P(G;" has a 1-factor) lim = l . n+m P( Gzn has no isolated vertices)

Remarks. Quite similarly to [2, p. 1601, if IV(G")ln'(l - P ~ ) ~ " - - - + const ZO, then the statement of Theorem 1 is not true ("cherries" can be ?f;;oth parts).

For strongly cubelike sequence condition (2) is equivalent to

+ 2?[ f ( n ) + In n ] l n , where f (n)- 00 . (2') > 1 - 2-0.5 n+m P f l -

Corollary 1 implies the validity of Weber's conjecture.

MATCHING IN RANDOM SPANNING SUBGRAPHS OF CUBELIKE GRAPHS 281

5. PRELIMINARY LEMMAS

Lemma 1. (Sapozhenko [S], Lemma 2.2; first version [7]). Let G be a connected graph on u vertices with maximal degree n. Then for any 1 5 i I u the number of i-vertex connected induced subgraphs of G is at most u(4n)'-'.

The proof of Lemma 4 of [l] proves in fact the following assertion.

Lemma 2. (Bollobh [l], Proof of Lemma 4; [2], Proof of Lemma 5 on p . 343). Let H be a graph with IV(H)( = u , A ( H ) 5 A and d =21E(H)(Iu, A + 1 5 x 5 u - A - 1 . Then

Lemma 3. Let H = (V, E ) be a connected graph, X C V , Z = V\[N,(X) U XI. Then there exists Y 3 X U Z such that I Yl I 31x1 + 212) and H( Y ) is connected.

Proof. We will construct sets Wi ( 1 5 i 5 ( X U Zl) such that H( v.) is connected, ) W i n ( X U Z ) ( = i a n d IWiViJ531WinXl+2(W,nZl. Takeanyy E X U Z a n d p u t W, = { y } . Suppose then that 2 5 i 5 ( X U Z( and WiPl is constructed. If there is uE(XUZ)flNH(Wi- , ) , then we put Wi=Wi-,U{u}. Let ( X U Z ) n

Z, then we put Wi = Wi-l U { w , u } . If w g X U Z, then w E NH(X) and there exists xEX\Wi-, such that ( x , w ) E E. In this case we put Wi= Wi-* U

Lemma 4. Let G(V, E ) be a cubelike graph of degree n and n I i I ( V ( /2 . Then the number of i-element sets A C V such that G(A) is connected and the inequality

&,(IyI:-l ) = 0, w E N#,( y- ,)hwi- 1, u E ivH( { w } ) n N ~ ( w,- ,). If w E x u

{x, w, u } .

IE,(A, V\A)/ 5 5i log n (3)

takes place is at most [l + 0 ( 1 ) ] V ( 2 ~ ' ( ' " ~ ~ n) 'n .

Proof. Let \ A ] = i , ]&(A, V\A)( 5 5i log n. Denote by H the skeleton of G ( A ) , i.e., the graph obtained from G ( A ) by substituting instead of every couple of multiple edges a single one. Then IV(H)I = i, A ( H ) 5 n and, due to condition (ii) of the definition of cubelike graphs,

Let x = [ i In n) In] . By Lemma 2,

6 log n IA\[X u NH(X)]l 5 (f ) i{ 1 - 1 + ___

{ X C A : I X I = x ) n i(ln n)(n + 1) I} + (I - +) exp[ - ni

5 ( f ) i[6(log n) /n + exp(-ln n)] 5 (f)7i(log n) ln .

282 KOSTOCHKA

Hence for at least (1 - 10g-O.~ n) ( 1 ) x-element sets X C A

lA\[X U NH(X)] ( 5 n)7i(log n)ln = 7i(log1.’ n)ln . (4)

Because

i - 1 i - 1 ( X C A : ( X I = x ) x - 1 x - 1

= ( X i ) 5 x ~ o g n ,

then there exists an x-element set XC A satisfying (4) and

By Lemma 3 there exists [A\N,(X)] C Y C A such that G ( Y ) is connected and

(YI 531x1 +2(A\(XU N H ( X ) ) ( I [ 1 + o(1)]14i(log1.’ n ) l n .

For sufficiently large n we may assume lY1 = y = L15i(log1.’ n ) / n ] . By Lemma 1 the total number of candidates for Y is at most IV1(4n)Y-1. Every such Y contains less than 2’ suitable X . Due to (5) for these Y and X we have I Y U NG(X)I I i{l + [ l + o(l)]5- (log2 n)/n}. Consequently, for large enough n, the number of required A’s does not exceed

[V1(8n)’( l i ( l + 6(log2n)/n]) I

Lemma 5. there exists x E X such that G\{x} is connected.

Let G ( X , Y; E) be a connected bipartite graph and let 1x1 I f Yl. Then

Proof. Suppose the Lemma is false and G = (X, Y; E) is an edge minimal counterexample. Then G is a tree and Y 3 { u E V(G): deg, u = l} = Vl(G). Let y E V l ( G ) and (x, y ) E E. If deg, x > 2 then G\{ y} contradicts the minimality of

def

C. If deg, x = 2 then G\{ y, x} contradicts the minimality of G.

6. M A I N LEMMA

Lemma 6. p 5 1. Then with probability at most 2- G , contains a pair (A, B) of sets satisfying the following conditions

Let G = ( X , Y ; E ) be a cubelike graph of degree n and 1 - 2-O.’ I the random spanning subgraph O S n [ l + o ( l ) ]

MATCHINCS IN RANDOM SPANNING SUBGRAPHS OF CUBELIKE GRAPHS 283

a. A C X , B C Y ; b. 1B1IlA1, 3 5 1 A ( , I B ( I ( Y 1 / 2 ; c . G ( A U B) is connected; d. NGp(A) C B.

Proof. Let a pair ( A , B) satisfy a-d and t = max(3, IBI}. By Lemma 3 there exists A' C A such that IA'I = t and the pair (A', B) also satisfies a-d. Since in our n-regular bipartite graph G we have \NG(A')l 2 [A'\ , then for IB( < t there exists B' 3 B such that IB'I = t and the pair (A', B') satisfies a-d, too. Hence we may replace the condition b by

So, the desired probability is no more than E!$"" P(gi), where gi is the event that G, contains a pair ( A , B) satisfying a, c, d and

b". IAI = IBI = i

Case 1 . 3 5 i 5 n/2 . By Lemma 1 the number of pairs ( A , B) satisfying condi- tions a, b" and c is at most IV1(4n)2i-'. Due to condition ii of the definition of cubelike graphs, we have IE,(A, YU?)lz i(n -log n - 1 ) . Hence

0.5(n- log n-1 ) 1 p ( g i ) 5 IV((4n)2'-'(1 - p)i("-log n - i ) 5 n2"( 16n2 2- 1 16nZ)' n l / i i / 2 + n / i 2-(n-logn)/2

- ( 2 I ( n 1 / 3 23/2+n/3 2 - n / Z n 1 / 2 1 6 ~ 2 ) ' = 2-in[l+o(1)]/6

Case 2. which satisfies a, b", c, d such that IEG(A, r \ B ) l I 5i log n. Then

n/2 < i 5 1x1 / 2 . Let '3; be the event that V(G,,) contains a pair (A, B)

Suppose 9 I takes place. Since

IE,(A U B , V\(A U B))I = 21EG(A, Y\B)I I (2Q.5 log n ,

then by Lemma 4 and the definition of cubelike graphs

284 KOSTOCHKA

7. PROOF OF THEOREM 1

If a bipartite graph G = (8, Y ; k) has no matching covering all the vertices of X or of P, then by Konig's theorem there are nonempty X,, X,, Y , , and Y2 such

N G ( Y 2 ) C X 2 , IY21>/X21. Of course, in this case min{/X,l, IYII}<IXU Y l / 4 . Let G" = ( X , Y ; b) be an induced subgraph of G;" on the set of nonisolated

vertices of G;". As noted above, the probability of the event that G" has no matching which covers X or ? does not exceed P(Sx) + P($y), where Sx is the event that Gin contains XI C X", Y , C Y" such that

that XI U X , = 8, X, nX2=O= Y, n Y2, Y , u Y2= 9, \ X l ~ > ~ Y l ~ , N G ( X , ) c Y , ,

IXll > IY, lz 1 , IY,l< IX" u Y"1 /4 , (6)

and Sy is the event that Gin contains Y, C Y", X, C X" such that

I Y21 > IX,l 2 1 , IX,l < IX" u Y" 1 /4 ,

where G = Gin. Let us estimate P(Sx). Note that if Sx takes place than there exists a minimal

(relative to IX, U Yl l ) pair (XI, Y,) satisfying (6)-(8). Because of this minimality the pair ( X , , Y , ) should satisfy not only (6)-(8) but also

GZn((X1 U Y l ) is connected,

1x11 = 1 + lYll .

(9)

(10)

MATCHINCS IN RANDOM SPANNING SUBCRAPHS OF CUBELIKE GRAPHS 285

Let 92 and 9; be the events that there exist Xl C X", Yl C Y" which satisfy (7)-( 10) and in addition

1x11 = 2 (the event 9;) ,

lXl[?3, IY,l<lV(G")I/4 (theevent Sl;).

Then P(?Fx) 5 P(94) + P(9:). In fact 9; means that there are xl, x2 E X" and y E Y" such that NG( { x , } ) = NG( {x2}) = { y } . Consequently,

According to (2), the last expression tends to 0 as n - , ~ . By Lemma 6 , P(9j;) = o(1). So, P(.Fx)-o(l). Analogously, P(FY) = o(1).

Remark. The proof implies that Theorem 1 will remain true after some weaken- ing conditions iv and ii in the definition of cubelike graphs. 0

ACKNOWLEDGMENTS

The author thanks K. Weber for many useful discussions and A. Rucinski for valuable remarks.

REFERENCES

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( 1979). [6] A. V. Kostochka, Maximum matchings and connected components of random span-

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[7] A. A. Sapozhenko, Metric properties of almost all Boolean functions, Discrete Anal., 10, 91-120 (1967) (in Russian).

[8] A. A. Sapozhenko, On the number of connected subsets with given cardinality of boundaries in bipartite graphs, in Methods Discrete Anal. Solving Combinat. Prob- lems, 45, pp. 42-70 (1987) (in Russian).

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[lo] K. Weber, On the evolution of random graphs in the n-cube, in Graphs, Hypergraphs and Applications, Proc. Conf. Eyba, DDR, 1984, Teubner Verlag, Leipzig, 1985, pp. 203-206.

Received October 4, 1989; revised January 25, 1990