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Uniform Distribution of a Subgraph in a Graph
Annals of Discrete Mathematics 17 (1983) 657-664 @ North-Holland Publishing Company
UNIFORM DISTRIBUTION OF A SUBGRAPH IN A GRAPH Thomas ZASLAVSKY * Deparimeni of Mathematics, The Ohio Siate University, Columbus, Ohio 43210, USA
1. Introduction Given a graph M of order r (perhaps having multiple edges but not loops), we discuss the property of a graph G that M is uniformly distributed in it, that is every induced subgraph of G of order r contains the same number of subgraphs isomorphic to M. It turns out that the order of G is bounded unless M = E,, or M is not contained in G at all, or G = pKn (K,, with multiplicity p on every edge), and quite strictly so if M and G are simple, Indeed in the latter case, if G 2 A4 and Gf Kn then n = I V(G)IG 2r - 2, and I suspect the true bound is n s r + 2. But the exact bound when G or M may be a multigraph is very much an unsolved problem. Background I became interested in uniform distribution of a subgraph because of the problem of complementation identities for generalized matchings. A generalized I-matching with model M is a disjoint union of 1 copies of M; it is an ordinary matching if M is an edge. For the generalized matching polynomial of a connected model, that is a M ( G;x ) =
'30
( - I)'m, (M; G)x"-'',
where ml (M; G ) is the number of generalized 1-matchings in G, there is the complementation identity a M ( G ; x ) =rz.0 * i ( M ; G ) a M ( p X n -;x), rj
(1)
where G p K . and m, (M; G ) is the number of generalized i-matchings in pKn of which no constituent M lies in G. What is remarkable about (1) is that the Research supported by the National Science Foundation 657
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coefficients in the expansion of a ’ ( G ; x ) in terms of the a M ( p K N; x ) have a simple combinatorial interpretation. (The proof is also simple.) What properties explain this? (Identity (1) was observed first by Godsil[3] for matchings and then generalized in [5].) Looking for an answer led me to consider the Appell property, D p N ( x )= N p N - , ( x and ) deg(pN(x)) = N, which the sequence Q “ ( p K N;x ) has. It is an open question whether or not an Appell sequence of matching polynomials must have an analog of (1); there is no obvious connection, so the answer probably depends on which such sequences exist. But there is an obvious combinatorial property of Appell sequences [5]. The associated graphs GN have uniform expected distribution of generalized matchings modelled on M: for each I 3 0, the average number of generalized I-matchings in induced subgraphs of G, of order rl is a function of 1, independent of N (provided N b rl). Therefore one asks: Which graph sequences have uniform expected distribution? That seemed quite difficult, so I took the strengthened condition of uniform distribution of generalized matchings: the number of generalized l-matchings in GN on any set of rl vertices should be the same. This is equivalent to uniform distribution of M alone. Thus by Theorem 1 of this paper the only solutions for E ( M )# 0 are GN = pKN and sequences of graphs not containing M. In the sequel we will always have the following situation: M is a graph of order r, not g,. G is a graph of order n b r in which M is uniformly distributed (except in Coroliary 9), but not a trivial example, which means we assume n > r and G 2 M but G # p K n .(Thus r b 3.) m ( M , G ) means the number of copies of M on each r-set X C V(G). We write g, for the graph without edges. For X C V(G), X‘ is its cornplement. X will also denote the subgraph of G induced by X , as in m (M; X). Adjacency is indicated by . The graph u + H is H with an extra vertex u adjacent to every vertex.
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2. The Ramsey bound We begin by observing that Ramsey’s theorem implies an upper bound on the order of G. Theorem 1. The order of a nontrivial example G of any multiplicity is bounded by a function of r.
Proof. Let m = m ( M , G ) and choose p o so that
mo = m (MIp o K , ) m < m (M, (P O+ 1)K).
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Let p(x, y ) = the multiplicity of the edge xy in G.Color the pairs of vertices of G in three colors: black if p(xy) > po, red if p ( x y ) = po and mo = m, white otherwise. By Ramsey's theorem, if n > f ( r ) there is a homogeneous set of r vertices for white or black or of 2r - 3 vertices for red. But a homogeneous white or black r-set has the wrong number of copies of M (too few for white, too many for black), so there must be a homogeneous red (2r - 3)-set. That entails m = mo. Now let n > f ( r ) , let X be a largest homogeneous red set in V(G), and let y E x'if X # V ( G ) .We can certainly find W C X of size r - 1 such that either all p ( w y ) s poand some p ( w y ) < po,or all p ( w y ) s po and some p ( w y ) > po. In the former case m ( M , W U y ) < mo = m, a contradiction. Similarly for the other case. So G is homogeneous for red; that is, G = poK,.
3. Isolated vertices
If M has isolated vertices, we can in general neglect them, thereby making r smaller and improving the bound on n. The justification is the following result. (The example M = K2 U K , and G = K 2 U K 2 shows that one cannot always discard all the isolated nodes.) Proposition 2. Let M have s isolated vertices. If s 3 2r - n, and in particular if n 2 21 - 1, then M less its isolated ueitices is uniformly distributed in G.
I
Proof. We write M = M'U Rs. Define f ( S T ) for disjoint S, T c V(G) to be the number of M'in G using all the vertices of S and none from T. First we show f(S 7')= q ( i , j ) ,where IS I = i and IT1 = j, provided i + j S n - r. That is true for i = 0. (For j = n - r we use the hypothesis and for j < n - r an easy summation argument.) Since
I
I
I
1
f ( u~x T ) = f ( ~7)- f ( ~T
u x ) = q ( i , j )- q ( i , j + 1)
for xk2 S U T and i + j < n - r, we reach the conclusion by induction on i . N o w l e t j = O a n d i = r - s , i f r - s s n - r . Wehave f(SIO)=constantfor I S I = r - s. But this means precisely that M' is uniformly distributed. If n 5 2r - 1, then n 3 2 r - s for any s > 0. So we are entitled to assume henceforth that the number of isolated vertices in M is 0 if n 2 2r - 1, at most 2r - n - 1 otherwise. In particular, if n 3 2r - 1, there can be no r vertices with an isolated member in G. Hence we have the following.
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Corollary 3. If n s 2 r - 1, then each y E V(G) has a t least n neighbors.
-r
+1
We can deduce an easy bound for simple G.
CorollaEy 4. If M and G are simple, then n c (r - 1)'. Proof. The idea is to take a vertex x1 of degree 3 n - r + 1, in its neighborhood an xz of degree>[n - ( r -l)]- r + 1, and so on, thus building a clique xI,. . . ,x , - ~ .If n 3 r(r - 2 ) + 2, there will be at least r more vertices adjacent to ~ , particular an adjacent pair x , - ~ ,n,. This gives an r-clique, all of xl,. . . ,x ~ - in contradicting the nontriviality of G. 0 4. Two small examples The first example has r = 4 and M = a two-edge matching. Assuming G is simple, one can prove that n 6 6 = r + 2, the only G with order n = 6 is the octahedral graph CP6, and there are only three possible G with n = 5 (the circuit C,,the bowtie z M, and the pyramid z + C4).The arguments are omitted because they are rather tedious. A second example in which n = r + 2 (suggested by Lovasz) has r = 8, M = a 4-edge matching, and n = 10, G = the Petersen graph. One can check uniform distribution easily because G and its complement are edge-transitive, so there are only two kinds of induced r-subgraphs.
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5. Centers A center in a vertex set X is a vertex adjacent to every other in X . An important example is that in which M has a center. For instance, M may be the star S,-,. This example introduces the technique we use later to get our best bound for simple graphs. Let CP, denote the cocktail party graph on m vertices ( K , less a perfect matching). Proposition 5. Suppose M is simple and has a center u, and suppose G is simple. Then n = r + 1, r is odd, M \ u C CP,-I, and G = CP,+I.
Proof. Say M = u + MI.We write X i for a set of i vertices of G. Take an X , and a center y in it; let X,-, = X , \ y . Now let y ' E X , and consider X : = X , - , U y r . Any copy of M on X , (similarly on X : ) arises in one of the following ways:
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(i) MI lies in X,-l and u corresponds to y ; (ii) MI contains y and u goes to w E X , - , . A copy of M in X:of either type corresponds to one copy on X, by exchanging y for y ' , because y is a center. (We rely on the simplicity of M and G.)But if y ' is not a center in X i , a copy of type (i) on X, has no corresponding copy on X:. This, if it occurred, would violate uniform distribution. But type (i) does occur, since each w in type (ii) is a center for X, and can therefore be interchanged with y. We conclude that every y ' E X L is adjacent to all X,-l. In fact, X L is a coclique. Suppose not, say y y ' . Let w be a noncenter in X,-l. (It exists because X , cannot be a clique, since G is simple.) M has no copy on X , with u corresponding to w ; but if we replace w by y then it does. Since that contradicts uniform distribution of M, y and y f could not have been adjacent. Since n > r, we have I X:-l I 3 2. We show X,-, has no centers. Any such center z is also a center of X,, whence XfU z is a coclique; but that contradicts the adjacency of every y f E XC to every member of X,-l. (This argument demonstrates that the center in every X : is unique.) Finally let X:= (X,-l U yy')\x for x E X,-, and let z be the center of X:.We see that X = (X:\z)c is a coclique adjacent to all X:\z, in particular to y . If Xfr l Z # 0, there is a contradiction. So n = r + 1. Moreover by choosing x to be in turn we find that G consists of cocliques of size 2 and all each member of X,-l points in different cocliques are adjacent. 0
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6. Complete models If M is complete with constant multiplicity we have an easy strong bound on n.
Proposition 6. If M = AK,, then n = r + 1.
Proof. Since G is nontrivial, all edge multiplicities p ( x y ) are at least A. Uniform distribution means that
(rn is a constant), for every X , C V ( G ) . Multiplying the expressions for rn(M,X,) over all X, in a fixed X,+lwe get rn'"
=
fl (p ( AI ) w )) '-'= f(X,+,Y-',
uw s x, + I
where f ( Y ) is the number of copies of M in Y C V ( G ) .Thus, for any X , and
vex,,
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a constant. If n 3 r + 2, X,-, 5 V(G), and u, w, ye X ,-l, we can conclude by comparing X , = X,-, U w and X,-, U y that
whence p ( u w ) = p ( u y ) for immediate. 0
any
u, w, y E V(G).
The
proposition
is
There are no nontrivial examples of order n = r + 1 if r = 2, but for r 3 3 we can take, for instance, G = U p ‘ H , where p 2 A and H is any regular simple graph of order r + 1 besides K,+,and K,+l.One can describe all possible G in a fairly effective but complicated way.
7. Our best bound
We can improve greatly on Corollary 4 by an argument developed jointly with Paul Seymour.
Theorem 7 (Seymour and Zaslavsky). If M and G are simple, then n C 2r - 2. This is surely not the best bound. I conjecture n d r + 2. The two small examples given earlier show that n $ r + 1 in general, but they may be exceptional. Examples with n = r + 1, however, are plentiful. Lovisz pointed out to me that one can take G to be vertex transitive and A4 any spanning subgraph of its vertex deletion (so generalizing the example of Proposition 5). The theorem follows directly from Corollary 3 and the following lemma.
Lemma 8. If M and G are simple, then G has maximum degree at most r, at most r - 1 if n > 2 r - 2 . Proof. Again let Xistand for any i-set in V(G). Suppose z is a center in X,+l.Let X,= Xr+t\z. Call an edge p q ‘real’ if there is a copy of M in X,+, using it. We will show that every edge zy is ‘real’. If not, say zy is not ‘real’. Each copy of M \ u (where u E V ( M ) )in X,\ y extends to a copy of M in Xr+,\ y, therefore (by uniform distribution) to a copy in X,.So if zx is ‘real’, say with the correspondences u -+ z and u‘-+ x (where u, U‘ E V ( M ) ) ,
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then so is y x with v + y and v ’ + x . But now we have acopyof M \ u ’ o n X,\x, which extends to a copy of M on X I + \, x by v ’ + z. So zy is ‘real’ after all. Next suppose w $Z Xr+land x E X,. Since zx is ‘real’, there is a copy Mo of M using zx, on vertices X , + ,\ y say. Let X : = (Xr\ y ) U w. Any copy of M on X : (say with v -+ w ) yields a copy on X,+,\y (by shifting v to z), so by uniform distribution there are no other copies of M on X,+,\y. That means M , corresponds to a copy on X:using wx. Since x was arbitrary in X,, we conclude w x for all W EX,+, and x E X,. Now suppose z can be chosen with dG( z )3 r + 1 and let X,+2consist of z and r + 1 neighbors. By the preceding, X,+*\z is a clique; thus G = K,. But this we excluded as trivial. So every vertex in G must have d e g r e e s r. Let us return to the case where z is a center for X , + ] .We showed that every w P XI+] is adjacent to all of X , but not to z , hence XS is a coclique of size n - r. That entails n - r S a ( M ) ,the independence number of M. But a ( M )< r since M # K,.So n > 2r - 2 implies that there is a vertex in G with at least r neighbors (by Corollary 3). Thus a ( M ) S n - r k r - 1. The case a ( M ) = r is trivial. If a ( M )= r - 1, M is a star v + since there are no isolated vertices (assumption justified by Proposition 2), and by Proposition 5 we have n s r + 1, a contradiction. 0
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I suspect the restriction n > 2r - 2 in the lemma is removable, with the exception of M = K , U K2, where G = CP, has degree 4 = r and order n = 6 = 2r - 2. 8. Wide uniform distribution
Suppose each induced subgraph of G of order q contains the same number of copies of M, where 4 is a fixed number 5 r. Let us call this wide uniform distribution of M . The smallest possible 4 is the width of the distribution. (This idea was proposed to me by Erdos.) We can apply all the previous results with and obtain the properties of uniform distribution for a model of model M U Kq-, order q. But by Proposition 2, if n 5 4 + r then M is uniformly distributed. So there is no need to consider wide distribution separately except for small n. As an application we generalize a result of Sir66 141 (for simple G) and Boshk [1,2] - the case r = 2 of the following consequence of Proposition 6. Corollary 9. Suppose every q vertices of G support the same positive number of copies of K . Then G = p K n unless G has order n < q + r.
For r = 2 the nontrivial G are the regular graphs of order q + 1. An example
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of order n = q 1 valid for any r between 3 and q is any vertex-transitive multigraph (except p K , ) that contains an r-clique. I do not know any nontrivial examples of higher order with width q > r 3 3.
References [I] J. Boslk, Induced subgraphs, in: Proc. 6th Hungarian Coll. on Combinatorics, Eger (1981) submitted. [2] J. Bosak, Induced subgraphs with the same order and size, Math. Slovaca, submitted. [3] C.D. Godsil, Hermite polynomials and a duality relation for the matchings polynomial, Combinatorica 1 (1981) 257-262. [4]J. Sirlh, On graphs containing many subgraphs with the same number of edges, Math. Slovaca 30 (1980) 267-268. [S] T. Zaslavsky, Generalized matchings and generalized Hermite polynomials, in: Proc. 6th Hungarian Coll. on Combinatorics, Eger (1981), submitted.
UNIFORM DISTRIBUTION OF A SUBGRAPH IN A GRAPH Thomas ZASLAVSKY * Deparimeni of Mathematics, The Ohio Siate University, Columbus, Ohio 43210, USA
1. Introduction Given a graph M of order r (perhaps having multiple edges but not loops), we discuss the property of a graph G that M is uniformly distributed in it, that is every induced subgraph of G of order r contains the same number of subgraphs isomorphic to M. It turns out that the order of G is bounded unless M = E,, or M is not contained in G at all, or G = pKn (K,, with multiplicity p on every edge), and quite strictly so if M and G are simple, Indeed in the latter case, if G 2 A4 and Gf Kn then n = I V(G)IG 2r - 2, and I suspect the true bound is n s r + 2. But the exact bound when G or M may be a multigraph is very much an unsolved problem. Background I became interested in uniform distribution of a subgraph because of the problem of complementation identities for generalized matchings. A generalized I-matching with model M is a disjoint union of 1 copies of M; it is an ordinary matching if M is an edge. For the generalized matching polynomial of a connected model, that is a M ( G;x ) =
'30
( - I)'m, (M; G)x"-'',
where ml (M; G ) is the number of generalized 1-matchings in G, there is the complementation identity a M ( G ; x ) =rz.0 * i ( M ; G ) a M ( p X n -;x), rj
(1)
where G p K . and m, (M; G ) is the number of generalized i-matchings in pKn of which no constituent M lies in G. What is remarkable about (1) is that the Research supported by the National Science Foundation 657
Z Zaslavsky
658
coefficients in the expansion of a ’ ( G ; x ) in terms of the a M ( p K N; x ) have a simple combinatorial interpretation. (The proof is also simple.) What properties explain this? (Identity (1) was observed first by Godsil[3] for matchings and then generalized in [5].) Looking for an answer led me to consider the Appell property, D p N ( x )= N p N - , ( x and ) deg(pN(x)) = N, which the sequence Q “ ( p K N;x ) has. It is an open question whether or not an Appell sequence of matching polynomials must have an analog of (1); there is no obvious connection, so the answer probably depends on which such sequences exist. But there is an obvious combinatorial property of Appell sequences [5]. The associated graphs GN have uniform expected distribution of generalized matchings modelled on M: for each I 3 0, the average number of generalized I-matchings in induced subgraphs of G, of order rl is a function of 1, independent of N (provided N b rl). Therefore one asks: Which graph sequences have uniform expected distribution? That seemed quite difficult, so I took the strengthened condition of uniform distribution of generalized matchings: the number of generalized l-matchings in GN on any set of rl vertices should be the same. This is equivalent to uniform distribution of M alone. Thus by Theorem 1 of this paper the only solutions for E ( M )# 0 are GN = pKN and sequences of graphs not containing M. In the sequel we will always have the following situation: M is a graph of order r, not g,. G is a graph of order n b r in which M is uniformly distributed (except in Coroliary 9), but not a trivial example, which means we assume n > r and G 2 M but G # p K n .(Thus r b 3.) m ( M , G ) means the number of copies of M on each r-set X C V(G). We write g, for the graph without edges. For X C V(G), X‘ is its cornplement. X will also denote the subgraph of G induced by X , as in m (M; X). Adjacency is indicated by . The graph u + H is H with an extra vertex u adjacent to every vertex.
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2. The Ramsey bound We begin by observing that Ramsey’s theorem implies an upper bound on the order of G. Theorem 1. The order of a nontrivial example G of any multiplicity is bounded by a function of r.
Proof. Let m = m ( M , G ) and choose p o so that
mo = m (MIp o K , ) m < m (M, (P O+ 1)K).
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Let p(x, y ) = the multiplicity of the edge xy in G.Color the pairs of vertices of G in three colors: black if p(xy) > po, red if p ( x y ) = po and mo = m, white otherwise. By Ramsey's theorem, if n > f ( r ) there is a homogeneous set of r vertices for white or black or of 2r - 3 vertices for red. But a homogeneous white or black r-set has the wrong number of copies of M (too few for white, too many for black), so there must be a homogeneous red (2r - 3)-set. That entails m = mo. Now let n > f ( r ) , let X be a largest homogeneous red set in V(G), and let y E x'if X # V ( G ) .We can certainly find W C X of size r - 1 such that either all p ( w y ) s poand some p ( w y ) < po,or all p ( w y ) s po and some p ( w y ) > po. In the former case m ( M , W U y ) < mo = m, a contradiction. Similarly for the other case. So G is homogeneous for red; that is, G = poK,.
3. Isolated vertices
If M has isolated vertices, we can in general neglect them, thereby making r smaller and improving the bound on n. The justification is the following result. (The example M = K2 U K , and G = K 2 U K 2 shows that one cannot always discard all the isolated nodes.) Proposition 2. Let M have s isolated vertices. If s 3 2r - n, and in particular if n 2 21 - 1, then M less its isolated ueitices is uniformly distributed in G.
I
Proof. We write M = M'U Rs. Define f ( S T ) for disjoint S, T c V(G) to be the number of M'in G using all the vertices of S and none from T. First we show f(S 7')= q ( i , j ) ,where IS I = i and IT1 = j, provided i + j S n - r. That is true for i = 0. (For j = n - r we use the hypothesis and for j < n - r an easy summation argument.) Since
I
I
I
1
f ( u~x T ) = f ( ~7)- f ( ~T
u x ) = q ( i , j )- q ( i , j + 1)
for xk2 S U T and i + j < n - r, we reach the conclusion by induction on i . N o w l e t j = O a n d i = r - s , i f r - s s n - r . Wehave f(SIO)=constantfor I S I = r - s. But this means precisely that M' is uniformly distributed. If n 5 2r - 1, then n 3 2 r - s for any s > 0. So we are entitled to assume henceforth that the number of isolated vertices in M is 0 if n 2 2r - 1, at most 2r - n - 1 otherwise. In particular, if n 3 2r - 1, there can be no r vertices with an isolated member in G. Hence we have the following.
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Corollary 3. If n s 2 r - 1, then each y E V(G) has a t least n neighbors.
-r
+1
We can deduce an easy bound for simple G.
CorollaEy 4. If M and G are simple, then n c (r - 1)'. Proof. The idea is to take a vertex x1 of degree 3 n - r + 1, in its neighborhood an xz of degree>[n - ( r -l)]- r + 1, and so on, thus building a clique xI,. . . ,x , - ~ .If n 3 r(r - 2 ) + 2, there will be at least r more vertices adjacent to ~ , particular an adjacent pair x , - ~ ,n,. This gives an r-clique, all of xl,. . . ,x ~ - in contradicting the nontriviality of G. 0 4. Two small examples The first example has r = 4 and M = a two-edge matching. Assuming G is simple, one can prove that n 6 6 = r + 2, the only G with order n = 6 is the octahedral graph CP6, and there are only three possible G with n = 5 (the circuit C,,the bowtie z M, and the pyramid z + C4).The arguments are omitted because they are rather tedious. A second example in which n = r + 2 (suggested by Lovasz) has r = 8, M = a 4-edge matching, and n = 10, G = the Petersen graph. One can check uniform distribution easily because G and its complement are edge-transitive, so there are only two kinds of induced r-subgraphs.
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5. Centers A center in a vertex set X is a vertex adjacent to every other in X . An important example is that in which M has a center. For instance, M may be the star S,-,. This example introduces the technique we use later to get our best bound for simple graphs. Let CP, denote the cocktail party graph on m vertices ( K , less a perfect matching). Proposition 5. Suppose M is simple and has a center u, and suppose G is simple. Then n = r + 1, r is odd, M \ u C CP,-I, and G = CP,+I.
Proof. Say M = u + MI.We write X i for a set of i vertices of G. Take an X , and a center y in it; let X,-, = X , \ y . Now let y ' E X , and consider X : = X , - , U y r . Any copy of M on X , (similarly on X : ) arises in one of the following ways:
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(i) MI lies in X,-l and u corresponds to y ; (ii) MI contains y and u goes to w E X , - , . A copy of M in X:of either type corresponds to one copy on X, by exchanging y for y ' , because y is a center. (We rely on the simplicity of M and G.)But if y ' is not a center in X i , a copy of type (i) on X, has no corresponding copy on X:. This, if it occurred, would violate uniform distribution. But type (i) does occur, since each w in type (ii) is a center for X, and can therefore be interchanged with y. We conclude that every y ' E X L is adjacent to all X,-l. In fact, X L is a coclique. Suppose not, say y y ' . Let w be a noncenter in X,-l. (It exists because X , cannot be a clique, since G is simple.) M has no copy on X , with u corresponding to w ; but if we replace w by y then it does. Since that contradicts uniform distribution of M, y and y f could not have been adjacent. Since n > r, we have I X:-l I 3 2. We show X,-, has no centers. Any such center z is also a center of X,, whence XfU z is a coclique; but that contradicts the adjacency of every y f E XC to every member of X,-l. (This argument demonstrates that the center in every X : is unique.) Finally let X:= (X,-l U yy')\x for x E X,-, and let z be the center of X:.We see that X = (X:\z)c is a coclique adjacent to all X:\z, in particular to y . If Xfr l Z # 0, there is a contradiction. So n = r + 1. Moreover by choosing x to be in turn we find that G consists of cocliques of size 2 and all each member of X,-l points in different cocliques are adjacent. 0
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6. Complete models If M is complete with constant multiplicity we have an easy strong bound on n.
Proposition 6. If M = AK,, then n = r + 1.
Proof. Since G is nontrivial, all edge multiplicities p ( x y ) are at least A. Uniform distribution means that
(rn is a constant), for every X , C V ( G ) . Multiplying the expressions for rn(M,X,) over all X, in a fixed X,+lwe get rn'"
=
fl (p ( AI ) w )) '-'= f(X,+,Y-',
uw s x, + I
where f ( Y ) is the number of copies of M in Y C V ( G ) .Thus, for any X , and
vex,,
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a constant. If n 3 r + 2, X,-, 5 V(G), and u, w, ye X ,-l, we can conclude by comparing X , = X,-, U w and X,-, U y that
whence p ( u w ) = p ( u y ) for immediate. 0
any
u, w, y E V(G).
The
proposition
is
There are no nontrivial examples of order n = r + 1 if r = 2, but for r 3 3 we can take, for instance, G = U p ‘ H , where p 2 A and H is any regular simple graph of order r + 1 besides K,+,and K,+l.One can describe all possible G in a fairly effective but complicated way.
7. Our best bound
We can improve greatly on Corollary 4 by an argument developed jointly with Paul Seymour.
Theorem 7 (Seymour and Zaslavsky). If M and G are simple, then n C 2r - 2. This is surely not the best bound. I conjecture n d r + 2. The two small examples given earlier show that n $ r + 1 in general, but they may be exceptional. Examples with n = r + 1, however, are plentiful. Lovisz pointed out to me that one can take G to be vertex transitive and A4 any spanning subgraph of its vertex deletion (so generalizing the example of Proposition 5). The theorem follows directly from Corollary 3 and the following lemma.
Lemma 8. If M and G are simple, then G has maximum degree at most r, at most r - 1 if n > 2 r - 2 . Proof. Again let Xistand for any i-set in V(G). Suppose z is a center in X,+l.Let X,= Xr+t\z. Call an edge p q ‘real’ if there is a copy of M in X,+, using it. We will show that every edge zy is ‘real’. If not, say zy is not ‘real’. Each copy of M \ u (where u E V ( M ) )in X,\ y extends to a copy of M in Xr+,\ y, therefore (by uniform distribution) to a copy in X,.So if zx is ‘real’, say with the correspondences u -+ z and u‘-+ x (where u, U‘ E V ( M ) ) ,
Uniform distribution of a subgraph in a graph
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then so is y x with v + y and v ’ + x . But now we have acopyof M \ u ’ o n X,\x, which extends to a copy of M on X I + \, x by v ’ + z. So zy is ‘real’ after all. Next suppose w $Z Xr+land x E X,. Since zx is ‘real’, there is a copy Mo of M using zx, on vertices X , + ,\ y say. Let X : = (Xr\ y ) U w. Any copy of M on X : (say with v -+ w ) yields a copy on X,+,\y (by shifting v to z), so by uniform distribution there are no other copies of M on X,+,\y. That means M , corresponds to a copy on X:using wx. Since x was arbitrary in X,, we conclude w x for all W EX,+, and x E X,. Now suppose z can be chosen with dG( z )3 r + 1 and let X,+2consist of z and r + 1 neighbors. By the preceding, X,+*\z is a clique; thus G = K,. But this we excluded as trivial. So every vertex in G must have d e g r e e s r. Let us return to the case where z is a center for X , + ] .We showed that every w P XI+] is adjacent to all of X , but not to z , hence XS is a coclique of size n - r. That entails n - r S a ( M ) ,the independence number of M. But a ( M )< r since M # K,.So n > 2r - 2 implies that there is a vertex in G with at least r neighbors (by Corollary 3). Thus a ( M ) S n - r k r - 1. The case a ( M ) = r is trivial. If a ( M )= r - 1, M is a star v + since there are no isolated vertices (assumption justified by Proposition 2), and by Proposition 5 we have n s r + 1, a contradiction. 0
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I suspect the restriction n > 2r - 2 in the lemma is removable, with the exception of M = K , U K2, where G = CP, has degree 4 = r and order n = 6 = 2r - 2. 8. Wide uniform distribution
Suppose each induced subgraph of G of order q contains the same number of copies of M, where 4 is a fixed number 5 r. Let us call this wide uniform distribution of M . The smallest possible 4 is the width of the distribution. (This idea was proposed to me by Erdos.) We can apply all the previous results with and obtain the properties of uniform distribution for a model of model M U Kq-, order q. But by Proposition 2, if n 5 4 + r then M is uniformly distributed. So there is no need to consider wide distribution separately except for small n. As an application we generalize a result of Sir66 141 (for simple G) and Boshk [1,2] - the case r = 2 of the following consequence of Proposition 6. Corollary 9. Suppose every q vertices of G support the same positive number of copies of K . Then G = p K n unless G has order n < q + r.
For r = 2 the nontrivial G are the regular graphs of order q + 1. An example
T. Zaslavsky
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of order n = q 1 valid for any r between 3 and q is any vertex-transitive multigraph (except p K , ) that contains an r-clique. I do not know any nontrivial examples of higher order with width q > r 3 3.
References [I] J. Boslk, Induced subgraphs, in: Proc. 6th Hungarian Coll. on Combinatorics, Eger (1981) submitted. [2] J. Bosak, Induced subgraphs with the same order and size, Math. Slovaca, submitted. [3] C.D. Godsil, Hermite polynomials and a duality relation for the matchings polynomial, Combinatorica 1 (1981) 257-262. [4]J. Sirlh, On graphs containing many subgraphs with the same number of edges, Math. Slovaca 30 (1980) 267-268. [S] T. Zaslavsky, Generalized matchings and generalized Hermite polynomials, in: Proc. 6th Hungarian Coll. on Combinatorics, Eger (1981), submitted.