Pentagonal geometries with block sizes 3, 4 and 5

TL;DR

This paper constructs and extends the existence spectrum of pentagonal geometries with block sizes 3, 4, and 5, providing new examples and theoretical results on their structure and properties.

Contribution

It introduces a direct construction for an infinite sequence of pentagonal geometries with block size 3 and connected deficiency graphs, and presents numerous new geometries for block sizes 4 and 5, expanding the known existence spectrum.

Findings

Constructed an infinite sequence of pentagonal geometries with block size 3.

Presented 39 new geometries with block size 4 and 5 with connected deficiency graphs.

Extended the existence spectrum for PENT(4, r) and PENT(5, r) geometries.

Abstract

A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every line, or block, is incident with $k$ points, every point is incident with $r$ lines, and for each point $x$, there is a line incident with precisely those points that are not collinear with $x$. An opposite line pair in a pentagonal geometry consists of two parallel lines such that each point on one of the lines is not collinear with precisely those points on the other line. We give a direct construction for an infinite sequence of pentagonal geometries with block size 3 and connected deficiency graphs. Also we present 39 new pentagonal geometries with block size 4 and five with block size 5, all with connected deficiency graphs. Consequentially we determine the existence spectrum up to a few possible exceptions for PENT(4, $r$) that do not contain opposite line pairs and for PENT(4, $r$) with one opposite line pair. More generally, given $j$ we show that there exists a PENT(4, $r$) with $j$ opposite line pairs for all sufficiently large admissible $r$. Using some new group divisible designs with block size 5 (including types $2^{35}$, $2^{71}$ and $10^{23}$) we significantly extend the known existence spectrum for PENT(5, $r$).