In a general set-theoretic context, an independent set is defined as a set which avoids certain specified structures called blocks. A formula is given for the number of independent sets of cardinality $k$ in terms of the numbers of configurations (i.e. non-empty collections) of blocks

Forbes, Anthony D.

Grannell, Mike J.

Griggs, Terry S.

Czech digital mathematics library

On independent sets

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaAnthony D. Forbes; Mike J. Grannell; Terry S. GriggsOn independent setsMathematica Slovaca, Vol. 55 (2005), No. 4, 375--377Persistent URL: http://dml.cz/dmlcz/132953Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 2005Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMathematica Slovaca ©2005 . . , C l _ _ / o r k r > r x M . 0 - _ o-.-, Mathematical Institute Math. SlOVaCa, 55 (2005), NO. 4, 375-377 Slovák Academy of Sciences ON INDEPENDENT SETS A. D. F O R B E S — M. J. GRANNELL — T. S. G R I G G S (Communicated by Martin Skoviera ) ABSTRACT. In a general set-theoretic context, an independent set is defined as a set which avoids certain specified structures called blocks. A formula is given for the number of independent sets of cardinality k in terms of the numbers of configurations (i.e. non-empty collections) of blocks. This paper is concerned with a formula for counting sets which avoid certain structures. The formula was originally produced in the context of Steiner triple systems [1] but is capable of considerable generalization. The authors believe that the result may be of some wider interest. Accordingly, we here present a version of the result in a more general setting. The proof relies on an application of the inclusion-exclusion principle. Consider a finite set of points S with | 5 | = s. Define Sx = V(S) and, for i = 1,2, 3 , . . . , put Si+1 = SiUV(Si), where V(X) denotes the set of all subsets oo of the set X. Then put S = (J S{. We will say that a set X G S covers the point a G S if a G X0 G X1 G • • • G Xn = X for some sequence of sets XQ, Xx,..., Xn G S. If X G S, then the set {a G S : X covers a} will be called the foundation of X and denoted by X_. Let B = {B1,B2,..., Bm} be a fixed finite set of elements of S, each having a non-empty foundation; we will call the elements of B blocks. As an example of these definitions, we might take B to be a set of 4-cycles on the set S. Note that a 4-cycle (a, b, c, d) may be equated with the set {{a, b}, {b, c}, {c, d} , {d, a}} G S. The block B = (a, b, c, d) covers the points a,b,c,d G S and B_ = {a, b, c, d}. By a similar method, we may describe any undirected graph by listing its edges, and any directed graph by representing a directed edge (a, b) by {{a},{a,b}}. 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : Pr imary 05A19; Secondary 03E05, 05A05, 05B30. K e y w o r d s : independent set. 375 A. D. FORBES —- M. J. GRANNELL — T. S. GRIGGS An independent set in S is a subset of S which does not contain the foun­dation of any block B G B. A configuration X is any non-empty set of blocks. Two configurations Xx and X2 are called isomorphic if one may be obtained from the other by means of a bijective mapping from the points of X1 to the points of X2. We denote by b(X) the number of blocks in X and by p(X) the number of points in X . Continuing our earlier example, if a, b,... ,g are distinct points of S, then the configuration of two 4-cycles Xx = {(a, b, c, a7), (a, e, /, g)} is isomorphic to X2 = {(a, b, c, d),(b, e, / , # ) } , but not to K3 = {(a, b, c, d), (a, e ,c,a)} . We also have b(XJ = b(X>) = b(X3) = 2, and p(X x ) = p(X 2 ) = 7, p(X3) = 6. Next consider any k-element set W C S with k > 1. Denote by n(X, TV) the number of isomorphic copies of the configuration X with foundation in the set W. If there are exactly / blocks with foundation in W, then 1 = 0 => ^2(-l)b{x)n(X,W)=0, x i>i => ^(-D^)n(x, ^ ) = - [̂  - Q + Q + • • •+(-1)'-1 QI X = - 1 , where the sums extend over all isomorphism classes of configurations X with p(X) <k. It follows that the number of independent sets of cardinality k in S, denoted by Ik(S), is given by м*) = (?)+ £ £(-Dьw»(-ү,wo W V I Ш U Ł X |VV|=fc However, ^ ^ ( ^ , W) is evaluated by listing the k-element sets W C S and |W |=fc scoring + 1 for every copy of X with foundation in each such W. This is the same number as that found by taking each configuration X and extending its founda­tion in all possible ways to form a k-element subset of 5 , i.e. n(X, S)(l_(x\) • In consequence, we arrive at the following formula, which we state formally as a theorem. 376 ON INDEPENDENT SETS THEOREM 1. The number of independent sets of cardinality k > 1 in the set S is given by k-p(X)J' where s = \S\ and the summation extends over all isomorphism classes of con-figurations X with p(X) < k. In the case of a Steiner triple system on a point set S, the blocks are those triples of points which form the system. In this case the formula is efficacious for low values of k (k < 8) because it is possible to determine or to estimate the values of n(X, S) for configurations covering at most k points. By this method we were able to determine the spectrum of maximum independent set sizes and of chromatic numbers for Steiner triple systems of order 21 . Both of these results are presented in [1]. We offer the more general version of the formula here in the hope that it will prove useful in other contexts. We conclude by remarking that if B and B1 are collections of blocks whose elements have identical foundations (i.e. if for each B G B there is a B' G B1 such that B = L?', and vice-versa), then the independent sets corresponding to B are identical to those corresponding to B'. In other words, the number Ik(S) depends on the foundations of the blocks rather than on the blocks themselves. A consequence of this observation is that there is no need to consider the possibility of repeated blocks either in B or in the configurations X C. B. REFERENCES [1] FORBES, A. D.—GRANNELL, M. J .—GRIGGS, T. S.: Independent sets in Steiner triple systems, Ars Combin. 72 (2004), 161 169. Received February 10, 2003 Department of Pure Mathematics The Open University Walton Hall Milton Keynes MK7 6AA UNITED KINGDOM E-mail: tonyforbes@ltkz.demon.co.uk  m.j.grannell@open.ac.uk  t.s.griggs@open.ac.uk 377 

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Forbes, Anthony

Grannell, Mike

Griggs, Terry

Open Research Online

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On independent sets

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Institute of Mathematics AS CR, v. v. i.

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