Abstract: Publisher Summary This chapter focuses on sequential vertex colorings, where vertices are sequentially added to the portion of the graph already colored, and the new colorings are determined to include each newly adjoined vertex. Considerable literature in the field of graph theory has dealt with the coloring of graphs. The majority of this effort has been devoted to the theory of graph coloring, and relatively little study has been directed toward the design of efficient graph coloring procedures. Because numerous proofs of properties relevant to graph coloring are constructive, many coloring procedures are at least implicit in the theoretical development. The chapter describes the concept of sequential colorings is formalized and certain upper bounds on the minimum number of colors needed to color a graph, the chromatic number x(G). The chief results show that the recursive-smallest-vertex-degree-last-ordering-with-interchange coloring algorithm will color any planar graph in five or fewer colors. The algorithm is evidently quite efficient even on large planar graphs.