Sharp bounds for decompositions of graphs into completer-partite subgraphs

Sharp Bounds for Decompositions of Graphs into Complete r-Partite Subgraphs

David A. Gregory* Kevin N. Vander Meulent

DEPARTMENT O f M A THE M A TlCS AND STA TlSTlCS QUEEN’S UNIVERSITY

KINGSTON, ONTARIO, CANADA K7L 3N6

ABSTRACT

If G is a graph on n vertices and r 2 2, we let m,(G) denote the minimum number of complete multipartite subgraphs, with r or fewer parts, needed to partition the edge set, f(G). In determining m,(G), we may assume that no two vertices of G have the same neighbor set. For such reduced graphs G, w e prove that m,(G) 2 log,(n + r - l)/r. Furthermore, for each k 2 0 and r 2 2, there is a unique reduced graph G = G(r, k) with m,(G) = k for which equality holds. We conclude with a short proof of the known eigenvalue bound m,(G) 2 max{n+(G), n-(G)/(r - I)}, and show that equality holds if G = G(r, k). 0 1996 John Wiley & Sons, Inc.

1. INTRODUCTION

Throughout the paper, C will denote a graph with vertex set V(G), edge set E(G) , and order n = IV(G)l. For our purposes, graphs are simple; that is, they have no loops or multiple edges. Apart from this, our terminology follows that of Bondy and Murty [l]. An r-partite graph is one whose vertex set can be partitioned into r subsets (the vertex parts) so that each edge has its ends in different parts. A complete r-partite graph is an r-partite graph in which every two vertices from different parts are joined by one edge. A vertex part may be empty. Thus, for s 5 r , every complete s-partite graph is also complete r-partite. A complete multipartite graph is a graph that is complete r-partite for some r .

*Supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0005 134. +Current address: Department of Mathematics, Redeemer College, Ancaster, Ontario, Canada L9K 1 J4.

Journal of Graph Theory, Vol. 21, No. 4, 393-400 (1996) 0 1996 John Wiley & Sons, Inc. CCC 0364-9024/96/040393-08

394 JOURNAL OF GRAPH THEORY

Let nz(G), m*(G) denote the minimum number of complete multipartite subgraphs needed to partition, respectively cover, the edge set of G. The former was examined by Hoffman [8]. In this paper, we examine the following refinements of these parameters. (We are mainly interested in partitions, but results on covers are easily proved and are included as well.)

For r 2 2, let m,(G) and m:(G) denote the minimum number of complete r-partite subgraphs of G that are needed to partition, respectively cover, the edge set of G . We call these numbers the r-partite decomposition number and the r-partite cover number of G , respectively. Clearly, if s 5 r , then

with equalities holding throughout whenever s 2 w ( G ) , the order of the largest clique (complete subgraph) of G.

Example 1.1 [12,6]. complete graph of order n ) are, respectively,

The r-partite decomposition and r-partite cover numbers of K,, (the

n - 1 r - 1

m,(K,,) = 1-1 and m,*(K,) = [log, nl .

To see the first equality, partition n - 1 of the vertices into k = [(n - 1)/(r - I)] disjoint subsets Vj , 1 5 j 5 k , of size r - 1 or less. Now, for each j , let R,j be the complete r-partite subgraph that has each vertex of Vj as a singleton part, and all the vertices of K, not in any Vi, 1 5 i 5 j , as the remaining vertex part. Then each edge of K,, is in precisely one of the R j . Thus, m,(K,) 5 k = [(n - l ) / ( r - I)]. On the other hand, Graham and Pollak [4] (see also [I41 and Section 4) have shown that mz(K,,) = n - 1 . Also, it is easy to see that a complete r-partite graph can be decomposed into r - 1 (or fewer) complete bipartite graphs. Thus ( r - I)m,(K,) 2 m2(K,) = n - 1 and so equality holds. Harary, Hsu and Miller have observed [6, p. 132, A] that the minimum number of (not necessarily complete) r-partite graphs needed to cover the edge set of a graph G is precisely [log, ,y(G)1, where ,y(G) is the chromatic number. Since the r-partite graphs in a cover of K , may be taken to be complete, m;(K,,) = [log, nl.

Huang [9] has examined decompositions of K , into complete r-partite subgraphs that have precisely r nonempty vertex parts. He shows that, for most n , exactly [(n + r - 3 ) / ( r - 1)1 such subgraphs are needed, but that, for some n , such decompositions need not even exist. As we have defined them, r-partite decompositions and covers exist for all graphs G. In fact,

where P(G) is the minimum number of vertices needed to cover the edges of G. Since induced subgraphs of complete r-partite graphs are also complete r-partite, m,(G -

{x}) 5 m,(G) and m,*(G - { x } ) 5 m:(G) for each vertex x in G. Moreover, if there is a vertex y in G with the same neighbor set as x , then

We say that two vertices of a graph are similar if they have the same neighbor sets. (Thus, similar vertices cannot be adjacent.) We say that G is reduced if no two of its vertices are

SHARP BOUNDS FOR GRAPH DECOMPOSITION 395

similar. (The terms canonical [3, p. 181 and point determining [ 5 ] , have also been used for this notion.) From (2), we see that we may restrict our attention to reduced graphs when examining r-partite decomposition or cover numbers.

If R is complete r-partite, then vertices in the same vertex part are similar. Thus, m,(R) 5

m,(K,) and m:(R) 5 m:(K,), with equality in both cases if R has precisely r nonempty parts. Consequently, m,(G) 5 m,(K,)m,(G) and mT(G) 5 m,*(K,)m;(G) for all graphs G. By Example 1.1 this implies that for all r, s 2 2:

(3 ) r - 1 s - l

m,(G) 5 [ - lmr (G) and m,*(G) 5 [log,T rlml(G).

The complete graph K , gives equalities in (3).

sition and cover numbers to those for the bipartite case: Taking s = 2 in (1) and (3 ) , we obtain inequalities relating the general r-partite decompo-

m,(G) 5 m,(G) 5 (r - I ) m r ( G ) and m:(G) 5 m I ( G ) 5 [log, rlm:(G)

Decompositions of a graph by complete bipartite subgraphs have been examined by many authors; for an extensive list of references, see [ 131. Decompositions of a digraph by directed complete bipartite subgraphs are a special case since they may be regarded as decompositions of an associated bipartite graph by complete bipartite subgraphs. For recent work and references on the directed case see, for example, [7]. For a survey on decompositions by various classes of graphs, see [2].

Each cover (and so, each partition) of E(G) by complete r-partite subgraphs, R,, 1 5 , j 5 k , may be represented by an n X k incidence matrix D with entries from (0, 1 , . . . , r}:

p if the i th vertex is in the pLhvertex part of R, 0 if the i th vertex is not in RJ .

Using incidence matrices of decompositions by complete bipartite subgraphs, Hoffman [S] observed that if k = m,(G), then one can find at most 3!' vertices in G no two of which are similar. The results in the next two sections refine this inequality. Results on covers by complete r-partite subgraphs are easily obtained and are given first.

2. COVERS BY COMPLETE r-PARTITE SUBGRAPHS

A graph G is determined by the n X k incidence matrix D of any of its covers (or decompositions) by complete r-partite subgraphs: two vertices of G are adjacent if and only if the corresponding rows of D have distinct nonzero entries in some column. Thus, if G is reduced, then the rows of D must be distinct (but not conversely). Since each row of D has entries from (0, I , . . . , r}, we have the following inequality:

Theorem 2.1. (r + I ) ~ .

If a reduced graph G of order n has r-partite cover number k , then n 5

Theorem 2.1 implies that if G is reduced, then


The next theorem implies that for each r and k there is a unique reduced graph with mT(G) = k for which (the logarithm is an integer and) equality holds in (4).

Theorem 2.2 graph G[r, k] with r-partite cover number k and order n = ( r + l)k.

G[r, 01 is a single vertex for each r 2 2. For k 2 1, G[r, k] could only be the graph that has the full IZ = (r + l ) k by k matrix D[r, k] of all k-tuples of {0,1,. . . , I-} as the incidence matrix of an r-partite cover. Note that no two vertices of this graph are similar; for if x and y are two rows of D[r, k] that correspond to nonadjacent vertices and x, # y j , let w, z be the k-tuples with w; = 1, z j = 2, and all other entries zero. Then w or z is adjacent to just one of x or y as one of x;, y; is zero.

For each k 2 0 and r 1 2, there is (up to isomorphism) a unique reduced

Proof.

I

3. DECOMPOSITIONS BY COMPLETE r-PARTITE SUBGRAPHS

We say that two k-tuples, x = (XI, x2,. . . , xk) and y = ( y ~ , y2 , . . . , y k ) , are compatible if xi, yi are both nonzero and unequal for at most one i, 1 5 i 5 k. The rows of the incidence matrix of a partition of the edge set of a graph G must be pairwise compatible and, if G is reduced, they must also be distinct. If a collection of distinct and pairwise compatible k-tuples have common support (i.e., are nonzero on the same set of positions), then they must have distinct values at one support position and equal values at each of the others. In Theorem 3.1, we use this restriction to obtain a lower bound on the r-partite decomposition number.

Theorem 3.1. n 5 r2k - r + 1.

If a reduced graph G of order n has r-partite decomposition number k then

Proof. Let D be the incidence matrix of a partition of E(G) into k = m,(G) complete r-partite subgraphs. Then the rows of D are distinct and pairwise compatible k-tuples of {0,1,2,. . . , r}. Because there are 2k - 1 possible nonempty support sets and since at most r nonzero rows of D can have the same support, n 5 1 + r(2k - 1). I

Theorem 3.1 implies that if G is reduced, then

(In [8], Hoffman showed that mz(G) 2 log,(n).) The next theorem implies that for each r and k there is unique reduced graph G with m,(G) = k for which (the logarithm is an integer and) equality holds in (5).

Theorem 3.2. Given k 2 0 and r 2 2, there is (up to isomorphism) a unique reduced graph G( r , k) with r-partite decomposition number k and order n = r2k - r + 1.

First we define such a graph C ( r , k) for each r 2 2 and k 2 1. (Clearly, G(r, 0) =

K I for each r, so we may assume that k 2 1.) The vertex set V(G(r,k)) is formed by first taking the set of all k-tuples of (0, l } and then making r copies of each nonzero k-tuple by replacing the first 1 by each of the integers 1, 2, . . . , r. Regard two k-tuples as adjacent if they have different nonzero entries in some position. There are n = r(2k - 1) + 1 k-tuples and, since a number exceeding 1 can only be in the first nonzero position, there cannot be two positions where two of the k-tuples are both nonzero and different. Thus the k-tuples may be regarded as the rows of an incidence matrix of a decomposition by k complete r-partite graphs.

Proof.


Therefore, m , ( G ( r , k ) ) 5 k . By the same argument as that used on G[r, k ] in the previous section, G(r , k ) is reduced. Thus m,(G(r , k ) ) = k by Theorem 3.1.

It remains to establish the uniqueness. Suppose then that r and k are given and that G is a reduced graph with m,(G) = k and order n = r2k - r + 1. Let D be the incidence matrix of a decomposition of G by complete r-partite subgraphs Rj , 1 5 j 5 k . Since G is reduced, the rows of D are distinct. The vertices of G may be regarded as the rows of D , two vertices being adjacent if and only if the corresponding rows have different nonzero entries in some column. If k = 1, then D consists of a single column with the entries 0,. . . , r . Hence if k = 1, G = G ( r , 1). Suppose now that k 2 2.

Because D has II = r (2k - 1) + 1 rows, referring to the proof of Theorem 3.1, we see that for each of the possible 2k - 1 nonempty support sets, there must be r rows of D with that support set. In particular, for each 1 5 i 5 k , D must have r rows with support {i, i + 1 , . . . , k}. Taking the r rows of D with no zero entries (i = l) , we may relabel the vertex parts of each R i , and reindex the R ; (permute the columns of D ) , so that the first r rows of D are

D =

1 1 1 . . . 1 2 1 1 . . . 1

r 1 1 . . . 1

Then each of remaining rows of D has at most one entry greater than 1. Thus, by successively taking the sets of r rows of D with support {i, i + 1 , . . . , k } , i 2 2, and permuting the nonzero columns at each stage, we may assume that the first rk rows of D are:

D =

1 1 1 . . . 1

r 1 1 . . . 1

0 1 1 . . . 1

0 r 1 ... 1

0 0 ... 0 I

0 0 . . . 0 r


Suppose x is one of the remaining rows in D and x; > 1 . If there were a position j < i such that xJ # 0, then x would not be compatible with one of the rows having zeros only in the first j - 1 positions. Thus x must equal zero in the first i - 1 positions. Hence G = G ( r , k ) . 1

By partitioning the vertices of G ( r , k ) according to whether the last entry is zero, one, or greater than 1, we see that G ( r , k ) may be constructed from G ( r , k - 1) by first forming the composition G ( r , k - 1)[F2], [ l , p. 1081, and then joining each of the vertices of K,-1 to each of the vertices in one of the copies of G(r , k - 1) in the composition. Some examples for r = 2 and r = 3 are shown in Figure 1 .

4. EIGENVALUE BOUNDS FOR THE r-PARTITE DECOMPOSITION NUMBER

Let n+(G) = n + ( A ) and n-(G) = n - ( A ) denote the number of positive and negative eigenvalues, respectively, of the adjacency matrix of G. The numbers n+(G) and n-(G) , like m,(G) and m:(G), are not changed if one of two similar vertices is deleted [3, p. 181.

H. S. Witsenhausen has shown that m2(G) 2 max{n+(G), n-(G)} [4, Lemma I] , while Hoffman [8] proves that m(G) = m,(G) 2 n+(G) , and attributes this bound to Graham and Witsenhausen. Using quadratic forms as in [4J, we offer a short proof of a combined result.

Theorem 4.1. For every graph G, m,(G) 2 max{n+(G),n-(G)/(r - 1)).

Let k = m,(G) and suppose that R I , R 2 , . . . , Rk are complete r-partite subgraphs that partition E(G) . Let XI, x 2 , . . . , x, be n indeterminants and let x = ( X I , x2, . . . , x,!)~. If A, is the n X n adjacency matrix of R , (with rows and columns indexed by vertices not in R , set equal to zero) and if X I , is the sum of the indeterminants corresponding to the vertices in the jth vertex part of R , (with X I , = 0 if the vertex part is empty), then

Proof.

Let x T A x = x ~ ( A I + . . . + Ak)x 5 0 whenever x E W . Note that dim(W) 2 n - k . Let U be the space of dimension n+(G) spanned by the eigenvectors of A corresponding to its positive eigenvalues. Then U n W = {0} since the quadratic form x T A x is positive definite on U . Thus n 2 dim(U) + dim(W) 2

n+(G) + n - k . Therefore k 2 n+(G). Now let W' = {x E R"lXij = 0, l 5 i 5 k , 1 5 j < r } and let U' be the space of di-

mension n-(G) spanned by the eigenvectors of A corresponding to its negative eigenvalues. Then dim(W') 2 n - k ( r - 1) and X'AX = 0 whenever x E W'. It follows as above that

W = {x E R " I x ; = l X i j = 0,I 5 i 5 k} . Then

(7 - l )k 2 n-(G). I Example 4.1. Because the adjacency matrix of K , has n - 1 negative eigenvalues (all equal to -I) , m,(K,) 2 [(n - l ) / ( r - I)]. This inequality was observed earlier in Example 1.1.

FIGURE 1


Note that, as in Example 1.1, the inequality ( r - l)rn,(G) 2 n-(G) in Theorem 4.1 can also be proved directly from the known inequality mz(G) 2 n-(G).

The two inequalities implicit in Theorem 4.1 are both best possible: the next theorem shows that the graph G ( r , k) found in Section 3 gives both equalities at once. We use the fact that if H is an induced subgraph of G then A(H) is a principal submatrix of A(G) and so by the interlacing theorem for eigenvalues [ 10, Sec. 4.31,

n + ( H ) 5 n+(G) and n - ( H ) 5 n-(G). (6)

Theorem 4.2. If G = G(r,k) , then n+(G) = k and n-(G) = k(r - 1).

Proof. Let H ( r , k) be the induced subgraph of G whose vertices are the k-tuples with exactly one nonzero entry. Since H( r ,k ) has k components, each a complete graph on r vertices, n + ( H ( r , k ) ) = k and n-(H(r,k)) = k(r - 1). Hence by (6), n+(G) 2 k and n- (G) 2 k(r - I ) . But m,(G) = k , so by Theorem 4.1, we have n+(G) = k and n-(G) =

k(r ~ 1). I

Unlike the bounds in Theorems 2.1 and 3.1, the bound in Theorem 4.1 may be attained by more than one reduced graph. In particular, Kartzke, Reznick. and West [ I I ] give families of graphs G for which mz(G) = max{n+(G),n-(G)}. They call such graphs eigensharp. Theorem 4.2 implies that the graphs G(2,k), k 2 0, are all eigensharp.

There is an eigenvalue lower bound for r-partite cover numbers that is analogous to Theorem 4.1. For every graph G ,

1 m:(G) 2 min max L ( M )

M (7)

where the minimum is taken over all graphs M with multiple edges that have underlying simple graph G, and the entries of the adjacency matrices A = A(M) count the number of edges between corresponding vertices. To see this, we first note that the term r-partite decomposition number is well-defined for graphs with multiple edges (but no loops) and that for every (simple) graph G,

m;(G) = min m,.(M)

where the minimum is taken over all graphs M with multiple edges that have underlying simple graph G. Inequality (7) then follows from this after observing that Theorem 4.1 also holds for graphs with multiple edges.

The graph G[r, k ] defined in Theorem 2.2 gives equality in (7). To see this, first note that the graph H ( r , k) in the proof of Theorem 4.2 is also an induced subgraph of G [ r , k]. In fact H ( 2 , k) is an induced matching in G[r, k ] . So any multigraph M with underlying graph G[r,k] contains an induced subgraph whose adjacency matrix is a direct sum of k positive multiples of A ( K 2 ) . Hence by (6), n + ( M ) 2 k for any M whose underlying graph is G[r, k ] . But m:(G[r,k]) = k, so we have equality in (7) for the graph G = G[r,k].

M

ACKNOWLEDGMENT

We are grateful to a referee for some suggestions for shortening the proofs of Theorems 3.1 and 3.2 and for some comments on notation.


Note added in pro08 author. They are reproduced there with the permission of John Wiley & Sons, Inc.

The results in this paper appear in the doctoral thesis of the second

References

[ 11 J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North Holland, New York (1976).

[2] F. R. K. Chung and R. L. Graham, Recent results in graph decompositions, Combinatorics, London Math. SOC. Lecture Note Ser. 52, Cambridge University press, New York (1981),

[3] D. CvetkoviC, M. Doob, I . Gutman, and A. TorgaSev, Recent Results in the Theory qf

[4] R.L. Graham and H.O. Pollak, On the addressing problem for loop switching, Bell

[5] D.P. Geoffroy and D.P. Sumner, An upper bound on the size of a largest clique in a

[6] F. Harary, D. Hsu, and Z. Miller, The biparticity of a graph, J. Graph Theory 1 (1977),

[7] K.A. Hefner, T.D. Henson, J .R. Lundgren, and J.S. Maybee, Biclique coverings of bigraphs and digraphs and minimum semiring ranks of (0, 1)-matrices, Congr. Num. 71

[8] A. J. Hoffman, Eigenvalues and partitionings of the edges of a graph, Linear Algebra

[9] Q. Huang, On the decomposition of K , into complete m-partite graphs, J. Graph Theory

[ 101 R. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York

[ l l ] T. Kratzke, B. Reznick, and D. West, Eigensharp graphs: decomposition into complete

[12] J. Liu and A. J. Schwenk, The number of multicolored trees in acyclic multiclique

[ 131 S. D. Monson, N. J. Pullman, and R. Rees, A survey of clique and biclique coverings and

1141 H. Tverberg, On the decomposition of K , into complete bipartite graphs, J . Graph Theory

100- 1 23.

Graph Spectra, North Holland, New York (1988).

System Tech. J. 50 (1971), 2495-2519.

graph, J. Graph Theory 2 (1978), 223-230.

13 1 - 133.

(1990), 115-122.

Appl. 5 (1972), 137-146.

15 (1991), 1-6.

(1985).

bipartite subgraphs, Trans. Amer. Math. Soc. 308, no. 2 (1988), 637-653.

decompositions, Congr. Num. 81 (1991), 129- 142.

factorizations of (0, 1)-matrices, Bulletin of the ICA. 14, (1995), 17-86.

6 (1982), 493-494.

Received February 1993

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Sharp bounds for decompositions of graphs into completer-partite subgraphs