Quasi-residual designs, \(1{1\over 2}\)-designs, and strongly regular multigraphs
Hall Jr. and Conner showed that every quasi-residual design with \(\lambda= 2\) is the residuum of a symmetric design, that is, it can be obtained from a symmetric design by deleting a line with all its points (notice that the same result for \(\lambda= 1\), just means that every affine plane has a projective closure). Even though the result is not true for \(\lambda> 2\), Bose et al. showed that there is a polynomial function \(f(\lambda)\) of degree 5 such that every quasi-residual 2- \((v,k,\lambda)\)-design is residual provided that \(k> f(\lambda)\). Neumaier improved the bound (by approximately a factor of \({1\over 2}\)) and showed that it suffices to demand \(k> {1\over 2}(\lambda^2- 1)(\lambda^3- \lambda^2-\lambda+ 2)\). In this paper, it is shown that a quasi-residual 2-\((v,k,\lambda)\)-design is the residuum of a symmetric design provided that \(k> c\lambda^4\) for a constant number \(c\). This result improves earlier results of Bose et al. (1976) and Neumaier (1982), who proved the result for \(k> {1\over 2} \lambda^5+ o(\lambda^4)\). This embedding theorem is a consequence of more general characterization theorems for certain strongly regular multigraphs. In order to simplify the proof, the author has not invested too much effort in finding the smallest possible value for \(c\). The author's results are the following theorems. Theorem 1: Any quasi-residual 2-\((v,k,\lambda)\)-design with \(k> \left({8\over\sqrt 3} \lambda+ \lambda+ 5\right)\lambda^2(\lambda- 1)\) is the residual of a symmetric design. Theorem 2: Suppose that \(T\) is a strongly regular multigraph whose parameters \((m, n, \mu, \nu, R)\) can be written in the form \(m= r\), \(n= k+ r+ c- 1- t\), \(\mu= rt\), \(\nu= re\), \(R= r(k- 1)\) with integers \(r\geq 3\) and \(t\geq 1\) and real numbers \(k> 0\) and \(c\geq 0\) if \(k> \left({8\over \sqrt 3} \nu+ \nu+ 5\right)\nu t\approx 5\), \(6n^2 t\), \(k> (c+ 1)t\) and \(r(c+ r- 1)\leq (r- 1)t\). Then \(T\) is the collinearity graph of a \(1{1\over 2}\)-design with parameters \((r, k, t, c)\).
scientific article; zbMATH DE number 790552