Section 2.3 Graph Coloring By Katie Lessard & Colleen Raimondi
a b c Section 2.3 Graph Coloring • Coloring- a coloring of a graph G assigns colors to the vertices of G so that adjacent vertices are given different colors. In the case of edge coloring, no edges that share a common vertex can be the same color. a b d c d
Section 2.3 Graph Coloring • Chromatic number- the minimal number of colors in a graph. a b d c The chromatic number for this graph is 3. Note: Each color class gives an independent set of vertices (vertices with no edges between them).
Section 2.3 Graph Coloring • Note: In order to verify that the chromatic number of a graph is a number k , we must also show that the graph can not be properly colored with k-1 colors. In other words the goal is to show that the (k-1)-coloring we might construct for the graph must force two adjacent vertices to have the same color.
Example 1 Pg. 77 in the text. • Find the chromatic number of the graph. The chromatic number is 4
Example 2 Pg. 78 in the text. • Find the chromatic number of the graph. This form of graph is called a wheel. The chromatic number is 4 In general, wheel graphs with an even number of “spokes” can be 3-colored whereas wheels with an odd number of “spokes” require 4 colors.
Example 3 Pg 79 in the text • Problem: A state legislature has many committees that meet for one hour each week. One wants a schedule of committee meetings times that minimize the total number of hours but such that two committees with overlapping membership do not meet at the same time. The chromatic number of this graph is four. Thus four hours suffice to schedule committee meetings without conflict.
Class Work Problem (Pg. 83 #2 b.) • For this graph give the chromatic color number for the vertices, the edges, and tell how this relates to bipartite graphs in general. Vertices colors = 2 Edge colors = 3 For bipartite graphs in general, the edge color is the the degree of the highest vertex, and the vertices color is always 2.