Incidence structures near configurations of type $(n_3)$

TL;DR

This paper explores a variant of $(n_3)$ configurations with specific line size constraints, investigates their planar representations, and extends Steinitz' theorem to this new setting.

Contribution

It introduces a new class of incidence structures with one line of size four and one of size two, and establishes their planar realizability and theoretical properties.

Findings

Existence of such configurations in planar representations is confirmed.

Steinitz' theorem is adapted for these modified configurations.

The study provides a classification framework for these structures.

Abstract

An $(n_3)$ configuration is an incidence structure equivalent to a linear hypergraph on $n$ vertices which is both 3-regular and 3-uniform. We investigate a variant in which one constraint, say 3-regularity, is present, and we allow exactly one line to have size four, exactly one line to have size two, and all other lines to have size three. In particular, we study planar (Euclidean or projective) representations, settling the existence question and adapting Steinitz' theorem for this setting.