Finite projective planes of even order with polarities (Q1312229)

Let \(\Pi = ({\mathcal P}, {\mathcal L},I)\) be an incidence structure consisting of a set of points \({\mathcal P}\), lines \({\mathcal L}\), and a relation \(I\) of incidence. If \(\pi\) is a polarity of \(\Pi\) (anti-automorphism of a partial plane \(\Pi)\), then a graph \({\mathcal G}\) can be associated with \(\Pi\) as follows: the vertices of \({\mathcal G}\) are pairs \((p,\pi (p))\), \(p \in {\mathcal P}\), and two vertices \((p, \pi (p))\), \((q, \pi (q))\) are joined by an edge if \(pI \pi (q)\) (and so \(qI \pi (p))\). The author exhibits in such a graph various properties of polarities of finite projective planes which were proved by \textit{R. Baer} [Bull. Am. Math. Soc. 52, 77-93 (1946; Zbl 0060.323)], and \textit{M. Seib} [Arch. Math. 21, 103-112 (1970; Zbl 0193.213)]. For example, a loop in the graph corresponds to an absolute point. While attempting to construct the graph of a finite projective plane of even order \(n\) with an orthogonal polarity, the author proves that such a plane exists if there exist certain permutations of the vertices of the graph.