Generalized pentagonal geometries. II (Q6611795)

In the last decades, many examples of partial linear spaces generalizing classical polygons were introduced, such as generalized quadrangles, generalized hexagons, or near-polygons.\N\NA generalized pentagonal geometry \(\mathrm{PENT}(k,r, w)\) 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. While \(w = k\), the structure is called a pentagonal geometry and denoted by \(\mathrm{PENT}(k,r)\). The classical pentagon is a \(\mathrm{PENT}(2, 2)\).\N\NThe deficiency graph of a \(\mathrm{PENT}(k,r, w)\) has as vertex set the set of points of the geometry, and there is an edge between \(x\) and \(y\) if and only if they are not collinear.\N\NIn this paper, the authors classify pentagonal geometries according to the properties of their deficiency graphs, and they investigate generalized pentagonal geometries \(\mathrm{PENT}(k,r, w)\) whose deficiency graph has girth 4. Moreover, they present some new constructions of \(\mathrm{PENT}(4,r)\) (e.g. \(\mathrm{PENT}(4,25)\)) and \(\mathrm{PENT}(5,r)\) whose deficiency graphs are connected. They prove that there exist pentagonal geometries \(\mathrm{PENT}(k,r)\) with deficiency graphs of girth at least 5 for \(r\geq13\), \(r \equiv1\pmod4\) when \(k = 4\), and for \(r\geq200000\), \(r\equiv0, 1 \pmod5\) when \(k = 5\).\N\NIn the last section, the authors define appropriate identifying codes for deficiency graphs of pentagonal geometries.\N\NFor Part I see [the authors, J. Comb. Des. 30, No. 1, 48--70 (2022; Zbl 1541.05021)].