An improved finiteness theorem for graphical t-designs

We prove that there exist only "nitely many nontrivial graphical t-(v; k; ) designs when k 6 4t=3. This improves a previous result of Betten et al. (Discrete Math. 197/198 (1999) 83–109). c © 2001 Elsevier Science B.V. All rights reserved. We use the notation and terminology of [1] and assume that the reader is familiar with the concept of graphical t-designs [2]. All polynomials in this note are polynomials in n. Betten et al. [1, Theorem 10] have shown that there exist only "nitely many nontrivial graphical t-(( 2 ); k; ) designs when k = t+1. In this note, we show that this "niteness result remains true when the condition k = t + 1 is relaxed to k 6 4t=3. Let t ¿ 3 and k 6 4t=3. Let I(t) denote the graph consisting of t independent edges and de"ne K to be the set of all graphs, each having k edges and contains I(t) as a subgraph. Then if G ∈ K, G must contain at least t=3 isolated edges. By Alltop’s Lemma (see [1, Lemma 2]), the entry in row G and column H of the polynomial Kramer–Mesner matrix is a polynomial whose degree is the di@erence in the sizes of the supports of G and H . Hence, the entry in row I(t) and column I(k) of the polynomial Kramer–Mesner matrix is a polynomial of degree 2(k − t). The other entries in row I(t) are polynomials of degree strictly less than 2(k − t). Without loss of generality, assume that I(k) is a block of a graphical t-(( 2 ); k; ) design D. The columns indexed by graphs in K\{I(k)} each has an entry a polynomial of degree 2(k − t), precisely in the row indexed by the graph obtained by removing k − t 6 t=3 isolated edges from the graph indexing the corresponding column. Hence for large n, all graphs in K\{I(k)} must also be blocks of D. This forces D to be the complete design and establishes the following result. ∗ Corresponding author. 12 Jambol Place, Singapore 119339, Singapore. E-mail address: [email protected] (Y.M. Chee). 0012-365X/01/$ see front matter c © 2001 Elsevier Science B.V. All rights reserved. PII: S0012 -365X(00)00059 -5 186 Y.M. Chee /Discrete Mathematics 237 (2001) 185–186 Theorem 1. There exist only /nitely many nontrivial graphical t-(( 2 ); k; ) designs when k 6 4t=3.