# Minimal Braces

## TL;DR

This paper introduces a new induction tool for minimal braces, establishing an upper size bound for minimal braces of order 2n and characterizing those meeting this bound.

## Contribution

It derives a main theorem for minimal braces based on McCuaig's brace generation theorem and applies it to characterize extremal minimal braces.

## Key findings

- Minimal braces of order 2n have size at most 5n-10 for n ≥ 6.
- Complete characterization of minimal braces that meet the size bound.
- Main theorem serves as an induction tool for minimal braces.

## Abstract

McCuaig (2001, Brace Generation, J. Graph Theory 38: 124-169) proved a generation theorem for braces, and used it as the principal induction tool to obtain a structural characterization of Pfaffian braces (2004, P{\'o}lya's Permanent Problem, Electronic J. Combinatorics 11: R79). A brace is minimal if deleting any edge results in a graph that is not a brace. From McCuaig's brace generation theorem, we derive our main theorem that may be viewed as an induction tool for minimal braces. As an application, we prove that a minimal brace of order $2n$ has size at most $5n-10$, when $n \geq 6$, and we provide a complete characterization of minimal braces that meet this upper bound. A similar work has already been done in the context of minimal bricks by Norine and Thomas (2006, Minimal Bricks, J. Combin. Theory Ser. B 96: 505-513) wherein they deduce the main result from the brick generation theorem due to the same authors (2007, Generating Bricks, J. Combin. Theory Ser. B 97: 769-817).