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MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 9, Friday, September 19

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 9, Friday, September 19. 2.4 Coloring Theorems. Homework (MATH 310#3F): Read 3.1. Write down a list of all newly introduced terms (printed in boldface or italic) Do Exercises 2.4: 2,4,8,10 Volunteers: ____________ ____________

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MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 9, Friday, September 19

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  1. MATH 310, FALL 2003(Combinatorial Problem Solving)Lecture 9, Friday, September 19

  2. 2.4 Coloring Theorems • Homework (MATH 310#3F): • Read 3.1. Write down a list of all newly introduced terms (printed in boldface or italic) • Do Exercises 2.4: 2,4,8,10 • Volunteers: • ____________ • ____________ • Problem: 2. • On Monday you will also turn in the list of all new terms (marked).

  3. Combinatorial “principles”. • So far we have encountered three methods that we call principles: • The bookkeeper’s principle • The induction principle • The addition principle • As we said earlier, we will learn several other useful principles.

  4. Triangulation of a Polygon. K A • A polygon is a plane circuit graph whose edges are drawn with straight lines. • By a triangulation of a polygon we mean the process of adding a set of straight-line chords between pairs of vertices of a polygon so that all interior regions of the graph are bounded by a triangle. • On the left we see a 12-gon. L B D J C I E H F G

  5. Triangulation of a Polygon. K A • A polygon is a plane circuit graph whose edges are drawn with straight lines. • By a triangulation of a polygon we mean the process of adding a set of straight-line chords between pairs of vertices of a polygon so that all interior regions of the graph are bounded by a triangle. • Here is a possible triangulation of a polygon. L B D J C I E H F G

  6. Theorem 1 K A • The vertices in a triangulation of a polygon can be 3-colored. • Proof. By induction on n, the number of edges of the polygon. • Hint: select a chord edge and partition the polygon into two smaller parts. • Note: The 3-coloring is unique. L B D J C I E H F G

  7. The Art Gallery Problem. K A • The Art Gallery has a shape of a polygon. • The Art Gallery problem: • What is the smallest number of guards needed to watch the paintings along the n walls? L B D J C I E H F G

  8. Corollary (Steve Fisk, 1978) K A • The Art Gallery problem with n walls requires at most b n/3 c guards. • Proof. Make a triangulation and color it with three colors. Take the least used color, say C,and place a guard at each corner colored C. L B D J C I E H F G

  9. Chromatic number • By c(G) we denote the chromatic number of G.

  10. Theorem 2 (Brooks, 1941) • If the graph G is not an odd circuit or a complete graph, then c(G) · d, where d is the maximum degree of G. • Note: there is an easy heuristic algorithm that will color any graph G with at most d+1 colors.

  11. Graph G and its Line Graph L(G) • Several exercises involve line graphs. On the left we se two graphs and the corrseponding line graphs.

  12. Coloring Edges • On the left we see an edge coloring of a graph. The minimum number of colors needed in such a coloring is called the edge chromatic number and is denoted byc’(G).

  13. Theorem 3 (Vizing, 1964) • If the maximum degree of a vertex in a graph G is d, then the edge chromatic number of G is either d or d+1. • In other words: d ·c’(G) · d+1.

  14. Theorem 4 • Every planar graph can be 5-colored.

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