MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 8, Wednesday, September 17

2.3 Graph Coloring Homework (MATH 310#3W): Read 2.4. Write down a list of all newly introduced terms (printed in boldface or italic) Do Exercises 2.3: 2,4,6,7,12,14,18 Volunteers: ____________ Problem: 7. On Monday you will also turn in the list of all new terms (marked). Tomorrow 3-5 in 201E – Office Hours

Coloring and Chromatic Number A coloring of a graph G assigns colors to the vertices of G so that adjacent vertices are given different colors. The minimal number of colors required to color a given graph is called the chromatic number of a graph.

Example 1: Simple Graph Coloring Find the chromatic number of the graph on the left.

Example 2: Coloring a Wheel Find the chromatic number of the graph on the left. Answer: 4.

Example 3: Unforced Coloring Find the chromatic number of the graph on the left. Answer: 4.

Example 4: Committee Scheduling There are 10 committees: A = {1,2,3,4} B = {1,6,7} C = {3,4,5} D = {2,4,7,8,9,10} E = {6,9,12,14} F = {5,8,11,13} G = {10,11,12,13,15,16} H = {14,15,17,19} I = {13,16,17,18} J = {18,19} Model graph with a vertex corresponding to each committee and with an edge joining two vertices if they represent committees with overlapping membership.

Example 4: Committee Scheduling There are 10 committees: A = {1,2,3,4} B = {1,6,7} C = {3,4,5} D = {2,4,7,8,9,10} E = {6,9,12,14} F = {5,8,11,13} G = {10,11,12,13,15,16} H = {14,15,17,19} I = {13,16,17,18} J = {18,19} How many hours are needed? B C D E F G H I J

Example 4: Committee Scheduling There are 10 committees: A = {1,2,3,4} B = {1,6,7} C = {3,4,5} D = {2,4,7,8,9,10} E = {6,9,12,14} F = {5,8,11,13} G = {10,11,12,13,15,16} H = {14,15,17,19} I = {13,16,17,18} J = {18,19} How many hours are needed? Answer: 4. B C D E F G H I J

Radio Frequency Assignment Problem In a given teritory there are n radio stations. Each one is determined by its position (x,y) and has a range radius r. The frequencies should be assigned in such a way that no two radio stations with overlaping hearing ranges are assigned the same frequency and that the total number of frequencies is minimal.

Radio Frequency Assignment Problem - Solution The problem is modeled by a graph G. Vertices of G are the circles centered at (x,y) with radius r. Two vertices are adjacent if the areas of the corresponding circles intersect. Frequencies are the colors. We are looking for an optimal coloring of G. The minimal number of frequencies is the chromatic number of G..

The Chromatic Polynomial Pk(G). The chromatic polynomial Pk(G) gives a formula for the number of ways to properly color G with k colors. The formula is polynomial in k.

Example 7: Chromatic Polynomial – Complete Graph Kn. Let us consider K4. Obviously P1(K4) = 0 P2(K4) = 0 P3(K4) = 0 P4(K4) = 4.3.2.1 P5(K4) = 5.4.3.2 P6(K4) = 6.5.4.3 P7(K4) = 7.6.5.4 In general: Pk(K4) = k.(k-1)(k-2).(k-3) More generally: Pk(Kn) = k.(k-1).(k-2) ... (k – n+1)

Chromatic Polynomial – Circuit C4. Pk(C4) = k(k-1)2 + k(k-1)(k-2)2. Here we used the Addition Principle (see also p.170): |A [ B| = |A| + |B|, if A Å B = ;.