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Non-symmetric configurations with deficiencies 1 and 2. (English) Zbl 0767.05034

Combinatorics ’90, Proc. Conf., Gaeta/Italy 1990, Ann. Discrete Math. 52, 227-239 (1992).
[For the entire collection see Zbl 0748.00018.]
A configuration is a finite incidence structure with \(v\) points and \(b\) lines (or blocks) such that each line contains \(k\) points, each point lies on \(r\) lines, two distinct lines meet in at most one point, and two distinct points are contained in at most one line. If \(v=b\) (and hence \(r=k\)), the configuration is called symmetric. Configurations with \(k=2\) are \(r\)-regular (simple) graphs on \(v\) vertices. The deficiency \(d\) of a configuration is defined by \(d=v-r(k-1)-1\). Thus each point is not connected to exactly \(d\) other points, and the deficiency is a measure of how much the configuration differs from a Steiner system.
The paper under review is a step towards showing the existence of a configuration with \(k=4\) and any admissible set of parameters \((vr=bk\), \(v\leq b\), and \(v\geq r(k-1)+1)\). The main emphasis is on configurations with small deficiency. Known results about divisible designs, regular packings of complete graphs and resolvable configurations are frequently used.

MSC:

05B30 Other designs, configurations

Citations:

Zbl 0748.00018
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