Coverage of random graphs in monochromatic components
Many problems of partitioning and covering the vertices of a graph are open, even though they have been present in the literature for more than 50 years. In this work, the interest is in delimiting how many monochromatic components are necessary to cover the vertices of a binomial random graph G(n, p) in any
edge coloring with r = 3 colors (tcr property). In the case of G(n, p), the main objective is to identify how different values of p are associated with the value of tc3 and to find thresholds for this property. This work presents some techniques that allow advances in relevant problems in Combinatorics, which are used in articles developed in the area. Continuing this research, we intend to further explore combinatorial techniques, investigate the partitioning property (tpr) of G(n, p) and the relationship of the independence number with the value of tcr.