On the Possibility of a Connection between the Construction of a Class of Bigeodetic Blocks and the Existence Problem for Biplanes

TL;DR

This paper explores the construction of bigeodetic blocks using combinatorial designs and investigates their potential connection to the existence problem for biplanes, bridging graph theory and combinatorics.

Contribution

It introduces two new classes of bigeodetic blocks constructed via BIBDs and suggests a novel link between graph constructions and the existence problem for biplanes.

Findings

Constructed two classes of bigeodetic blocks using BIBDs

Established a potential connection between graph theory and biplane existence

Linked the construction to the Bruck-Ryser-Chowla theorem

Abstract

Graph theory and enumerative combinatorics are two branches of mathematical sciences that have developed astonishingly over the past one hundred years. It is especially important to point out that graph theory employs combinatorial techniques to solve key problems of characterization, construction, enumeration and classification of an enormous set of different families of graphs. This paper describes the construction of two classes of bigeodetic blocks using balanced incomplete block designs (BIBDs). On the other hand, even though graph theory and combinatorics have a close relationship, the opposite problem, that is, considering certain graph constructions when solving problems of combinatorics is not common, but possible. The construction of the second class of bigeodetic blocks described in this paper represents an example of how graph theory could somehow give a clue to the description of a problem of existence in combinatorics. We refer to the problem of existence for biplanes. A connection between the mentioned construction, the Bruck-Ryser-Chowla theorem and the problem of existence for biplanes is considered.