Minimal braces

• Published in: Journal of graph theory Vol. 96; no. 4; pp. 490 - 509
• Main Authors: Fabres, Phelipe A., Kothari, Nishad, Carvalho, Marcelo H.
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01-03-2021
• Subjects: 2‐extendable graphs; bipartite graphs; braces; Hand tools; matching covered graphs; perfect matchings; Structural analysis; Theorems; Upper bounds
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.22628

McCuaig proved a generation theorem for braces, and used it as the principal induction tool to obtain a structural characterization of Pfaffian braces. 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≥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 wherein they deduce the main result from the brick generation theorem due to the same authors.