A version of sequential coloring algorithm is the smallest last approach (Matula, Marble,...

A version of sequential coloring algorithm is the smallest last approach (Matula, Marble, and Isaacson, 1972). In this version, a vertex with the lowest degree is always chosen and put at the front of the sequence of vertices. Afterward, the vertex is temporarily removed from the graph. In this way, if a vertex v is removed from the graph so is also each edge(vu), which lowers the degree of each neighbor of v. After the sequence of vertices is established, the links temporarily removed are restored and the for loop is executed to assign colors to vertices. Apply the smallest last approach to the graph in Figure 8a and find the upper bound for the chromatic number.

Figure (a) A graph used for coloring; (b) colors assigned to vertices with the sequential coloring algorithm that orders vertices by index number; (c) vertices are put in the largest first sequence; (d) graph coloring obtained with the Brélaz algorithm.