Polynomial Strategies for the 3-Coloring of a Graph

Abstract

The coloring of a graph is a problem of interest in the area of computer science due to the many applications it offers. The graph coloring problem has many utilities in areas like scheduling problems, frequency allocation, planning, etc. In the coloring of a graph, we want to color the nodes properly with the smallest possible number of colors. We present some necessary conditions for the 3-coloring of an input graph. All of those conditions can be checked in polynomial time. We also propose an appropriate combinatorial pattern representing proper 3-coloring of a graph based in its basic cycles, and where such pattern is codified via satisfy assignments of a two conjunctive Boolean formula. The Boolean formula is formed according to the basic cycles appearing in the graph. This paper shows the methodology for 3-coloring of a graph using 2-CF (Conjunctive form), as well as by several examples illustrate the calculation of it.