Classificação de grafos-junção quanto ao índice cromático

Abstract

The Classification Problem, which consists of finding the minimum number of colors in order to color every edge of a graph so that no adjacent edges receive the same color, is an NP-complete problem, although we have only two possibilities for that number, by Vizing’s Theorem. For specific classes of graphs, polynomial time algorithms have already been found. The class of the join graphs, a significantly populous class of graphs — it is a superclass of the cographs, complete graphs (Kn) and the complete multipartite graphs— does not have a polynomial-time algorithm for the Classification Problem. However, the overfull conjecture and a linear-time algorithm for recognizing overfull-subgraph graphs give us some evidences for the existence of a polynomial-time algorithm for the Classification Problem. Therefore, this work presents a study about the Classification Problem, aiming to propose some new results so that the state of the art can be improved.