1 / 14

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: ____________ ____________

gazit
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related