Autor(es)

Botler, F. ; Jimenez, A. ; Sambinelli, M. ; Wakabayashi, Y.

• Tipo de documento: Artigo Científico
• Fonte: GRAPHS AND COMBINATORICS; v. 40, n. 5, p. 21-pg., 2024-10-01.

Resumo

The 2-Decomposition Conjecture, equivalent to the 3-Decomposition Conjecture stated in 2011 by Hoffmann-Ostenhof, claims that every connected graph G with vertices of degree 2 and 3, for which G\E(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G \setminus E(C)$$\end{document} is disconnected for every cycle C, admits a decomposition into a spanning tree and a matching. In this work we present two main results focused on developing a strategy to prove the 2-Decomposition Conjecture. One of them is a list of structural properties of a minimum counterexample for this conjecture. Among those properties, we prove that a minimum counterexample has girth at least 5 and its vertices of degree 2 are at distance at least 3. Motivated by the class of smallest counterexamples, we show that the 2-Decomposition Conjecture holds for graphs whose vertices of degree 3 induce a collection of cacti in which each vertex belongs to a cycle. The core of the proof of this result may possibly be used in an inductive proof of the 2-Decomposition Conjecture based on a parameter that relates the number of vertices of degree 2 and 3 in a minimum counterexample. (AU)

• Processo FAPESP: 15/11937-9 - Investigação de problemas difíceis do ponto de vista algorítmico e estrutural
• Beneficiário: Flávio Keidi Miyazawa
• Modalidade de apoio: Auxílio à Pesquisa - Temático
• Processo FAPESP: 19/13364-7 - Problemas extremais e estruturais em teoria dos grafos
• Beneficiário: Cristina Gomes Fernandes
• Modalidade de apoio: Auxílio à Pesquisa - Regular