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Graph Coloring and Chromatic Number

Learn about graph coloring and the chromatic number of a graph, and solve combinatorial problems related to graph coloring. Explore examples and understand the concept of the chromatic polynomial.

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Graph Coloring and Chromatic Number

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

  2. 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

  3. 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.

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

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

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

  7. 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.

  8. Example 4: Committee Scheduling A • 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

  9. Example 4: Committee Scheduling A • 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

  10. 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.

  11. 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..

  12. 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.

  13. 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)

  14. 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 = ;.

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