The Projective Planarity Question for Matroids of $3$-Nets and Biased Graphs

TL;DR

This paper establishes algebraic criteria for embedding matroids derived from biased graphs and 3-nets into non-Desarguesian projective planes, linking graph theory, algebra, and geometry.

Contribution

It provides new algebraic conditions for realizing 3-nets and biased graphs in arbitrary projective planes, extending previous results beyond Desarguesian spaces.

Findings

Criteria depend on quasigroup embeddability into ternary rings

Every finite 3-node biased graph is a subgraph of a biased expansion of a triangle

Open problem: existence of finite quasigroups not embeddable in any finite ternary ring

Abstract

A biased graph is a graph with a class of selected circles ("cycles", "circuits"), called "balanced", such that no theta subgraph contains exactly two balanced circles. A biased graph has two natural matroids, the frame matroid and the lift matroid. A classical question in matroid theory is whether a matroid can be embedded in a projective geometry. There is no known general answer, but for matroids of biased graphs it is possible to give algebraic criteria. Zaslavsky has previously given such criteria for embeddability of biased-graphic matroids in Desarguesian projective spaces; in this paper we establish criteria for the remaining case, that is, embeddability in an arbitrary projective plane that is not necessarily Desarguesian. The criteria depend on the embeddability of a quasigroup associated to the graph into the additive or multiplicative loop of a ternary coordinate ring for the plane. A 3-node biased graph is equivalent to an abstract partial 3-net; thus, we have a new algebraic criterion for an abstract 3-net to be realized in a non-Desarguesian projective plane. We work in terms of a special kind of 3-node biased graph called a biased expansion of a triangle. Our results apply to all finite 3-node biased graphs because, as we prove, every such biased graph is a subgraph of a finite biased expansion of a triangle. A biased expansion of a triangle, in turn, is equivalent to an isostrophe class of quasigroups, which is equivalent to a $3$-net. Much is not known about embedding a quasigroup into a ternary ring, so we do not say our criteria are definitive. For instance, it is not even known whether there is a finite quasigroup that cannot be embedded in any finite ternary ring. If there is, then there is a finite rank-3 matroid (of the corresponding biased expansion) that cannot be embedded in any finite projective plane---a presently unsolved problem.