Coloração total de grafos bipartidos

Abstract

A total coloring in a graph G is a color assignment to elements to G so that any two adjacent elements have different colors. The Total Coloring Problem is to determine the smallest number of colors to obtain a total coloring for a given graph G. This number is called the total chromatic number and it is denoted by X"(G). In its decision version, the Total Coloring Problem receives as input a graph G and an integer number, k, and it is desired to know if there is a total coloring for G with at most k colors. The decision version of Total Coloring Problem is an NP - Complete problem, even when restricted to the class of bipartite graphs, which are those whose vertices can be partitioned into two independent sets. An efficient algorithm that solves any of the NP-Complete problems in polynomial time is unknown, but if any algorithm of polynomial complexity exists, then it can be used to solve any problem of the class NP. Presenting an efficient algorithm for an NP - complete problem or proving that such an algorithm does not exist is one of the seven most challenging problems of the millennium, according to the Clay Mathematics Institute. In this work we present a subset of bipartite graphs for which the Total Coloring Problem remains open.