Existence results for pentagonal geometries
Anthony D. Forbes, Terry S. Griggs, and Klara Stokes
TL;DR
This paper establishes new existence results for pentagonal geometries with block sizes 3 and 4, including complete spectra for certain configurations and partial results for others, advancing the understanding of these combinatorial structures.
Contribution
It provides the first complete existence spectra for PENT(3,r) with maximum and no opposite line pairs, and partial spectra for PENT(4,r), filling gaps in combinatorial design theory.
Findings
Complete existence spectra for PENT(3,r) with maximum opposite line pairs.
Complete existence spectra for PENT(3,r) without opposite line pairs.
Existence spectrum of PENT(4,r) with eleven possible exceptions.
Abstract
New results on pentagonal geometries PENT(k,r) with block sizes k = 3 or k = 4 are given. In particular we completely determine the existence spectra for PENT(3,r) systems with the maximum number of opposite line pairs as well as those without any opposite line pairs. A wide-ranging result about PENT(3,r) with any number of opposite line pairs is proved. We also determine the existence spectrum of PENT(4,r) systems with eleven possible exceptions.
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Taxonomy
TopicsMathematics and Applications · graph theory and CDMA systems · Finite Group Theory Research