Generalized pentagonal geometries -- II

TL;DR

This paper explores generalized pentagonal geometries, focusing on their deficiency graphs with girth 4, providing new constructions, existence theorems, and discussing identifying codes within these structures.

Contribution

It introduces new construction methods and existence results for generalized pentagonal geometries with specific deficiency graph properties, especially for girth 4 and higher.

Findings

Existence of PENT(3,r,w) with girth 4 deficiency graphs for various parameters.

Construction of new PENT(4,r) and PENT(5,r) geometries with connected deficiency graphs.

Existence of pentagonal geometries with deficiency graphs of girth at least 5 for large r and specific congruence conditions.

Abstract

A generalized pentagonal geometry PENT($k$,$r$,$w$) is a partial linear space, where every line is incident with $k$ points, every point is incident with $r$ lines, and for each point, $x$, the set of points not collinear with $x$ forms the point set of a Steiner system $S(2,k,w)$ whose blocks are lines of the geometry. If $w = k$, the structure is called a pentagonal geometry and denoted by PENT($k$,$r$). The deficiency graph of a PENT($k$,$r$,$w$) has as its vertices the points of the geometry, and there is an edge between $x$ and $y$ precisely when $x$ and $y$ are not collinear. Our primary objective is to investigate generalized pentagonal geometries PENT($k$,$r$,$w$) where the deficiency graph has girth 4. We describe some construction methods, including a procedure that preserves deficiency graph connectedness, and we prove a number of theorems regarding the existence spectra for $k = 3$ and various values of $w$. In addition, we present some new PENT(4,$r$) (including PENT(4,25)) and PENT(5,$r$) with connected deficiency graphs. Consequently, we prove that there exist pentagonal geometries PENT($k$,$r$) with deficiency graphs of girth at least 5 for $r \ge 13$, $r$ congruent to 1 modulo 4 when $k = 4$, and for $r \ge 200000$, $r$ congruent to 0 or 1 modulo 5 when $k = 5$. We conclude with a discussion of appropriately defined identifying codes for pentagonal geometries.