Acyclic edge-colouring of graphs

• Ano de defesa: 2025
• Autor(a) principal: Quispe Porras, Ariana Maite
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Acyclic chromatic index; Acyclic edge colouring; Coloração acíclica de arestas; Índice cromático acíclico
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/45/45134/tde-12012026-185219/

A proper edge-colouring of a graph G is called acyclic if there are no bichromatic cycles in G. The acyclic chromatic index of a simple graph G, denoted by a\'(G), is the smallest number k such that G admits an acyclic edge-colouring using k colours. Fiamik (1978) and Alon, Sudakov and Zaks (2001) independently conjectured that a\'(G) (G) + 2 for any simple graph G. This conjecture is well known as the Acyclic Edge-Colouring Conjecture (AECC). Currently, the best known upper bound for an arbitrary graph is 3.569((G) - 1), obtained by Fialho, de Lima and Procacci (2020). This was proved using a modification of the edge-colouring algorithm presented by Giotis, Kirousis, Psaromiligkos and Thilikos (2017). Furthermore, the AECC was proved for graphs with {3,4}, 2-degenerate graphs, 3-sparse graphs, some cases of complete graphs and some cases of planar graphs. The main objective of this work is to study the AECC for graphs with {3,4} and complete bipartite graphs K_{p,p} when p is prime. In particular, for the case = 3, we have an original and possibly more readable proof than those in published papers. Unfortunately, in the case = 4, we were only able to clean up some of the cases, since, again, the literature on this case is very difficult to read.