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Section 2.4

Tucker, Applied Combinatorics, Sec. 2.4, Prepared by Whitney and Cody. Section 2.4. Coloring Theorems.

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Section 2.4

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  1. Tucker, Applied Combinatorics, Sec. 2.4, Prepared by Whitney and Cody Section 2.4 Coloring Theorems Tucker, Applied Combinatorics, Sec 2.4

  2. Triangulation of a polygon: The process of adding a set of straight-line chords between pairs of vertices of a polygon so that all the interior regions of the graph are bounded by a triangle (these chords cannot cross each other nor can they cross the sides of the polygon). G Triangulation of G Chromatic number = 3 Definitions: Chromatic number: The smallest number of colors that can be used in a coloring of a graph Symbols: Let the symbol (G) denote the chromatic number of the graph G. Let the symbol r denote the largest integer  r. Tucker, Applied Combinatorics, Sec 2.4

  3. E E E Theorem 1: The vertices in a triangulation of a polygon can be 3-colored. PROOF: By induction • Let n represent the number of edges of a polygon. • For n=3, give each corner a different color. • Assume that any triangulated polygon with less than n boundary edges, n4, can be 3-colored and considered a triangulated polygon T with n boundary edges. • Pick a chord edge e, which split T into two smaller triangulated polygons, which can be 3-colored (by the induction assumption). • The two new subgraphs can be combined to yield a 3 coloring of the original polygon by making the end vertices of e the same color in both subgraphs. Tucker, Applied Combinatorics, Sec 2.4

  4. The Art Gallery Problem • The problem asks for the least number of guards needed to watch paintings along the n walls of the gallery. • The walls are assumed to form a polygon. • The guards need to have a direct line of sight to every point on the point on the walls. • A guard at a corner is assumed to be able to see the two walls that end at that corner. An application of Theorem 1: The art Gallery Problem with n walls requires at most n/3 Tucker, Applied Combinatorics, Sec 2.4

  5. Proof: • Make a triangulation of the polygon formed by the walls of the art gallery. • Make sure the guard at any corner of any triangle has all sides under surveillance. • Now obtain a 3-coloring of this triangulation. • Pick one of the colors (for example red) and put a guard on every red corner of the triangles. • Hence, the sides of all triangles, all the gallery walls, will be watched. • A polygon with n walls has n corners. • If there are n corners and 3 colors, some color is used at n/3 or fewer corners. Tucker, Applied Combinatorics, Sec 2.4

  6. Theorem 2 Brook’s Theorem: If the graph G is not an odd circuit or a complete graph, then (G)  d, where d is the maximum degree of a vertex of G. Tucker, Applied Combinatorics, Sec 2.4

  7. Theorem 3: For any positive integer k, there exists a triangle-free graph G with (G) = k. (ie. There are graphs with no complete subgraphs, that take many colors) Note: X(G)  N, where N is he size of the largest complete subgraph of G Tucker, Applied Combinatorics, Sec 2.4

  8. Instead of coloring vertices you color edges so that the edges with a common end vertex get different colors. • A very good bound on the edge chromatic number of a graph in terms of degree is possible. • All edges incident at a given vertex must have different colors, and so the maximum degree of a vertex in a graph is a lower bound on the edge chromatic number. • Even better, one can prove theorem 4… Tucker, Applied Combinatorics, Sec 2.4

  9. Theorem 4: Vizing’s Theorem 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. Tucker, Applied Combinatorics, Sec 2.4

  10. X X Theorem 5: It has already been proven that all planar graphs can be 4-colored but it is very long and complicated so lets move on to the next best thing…5-coloring Every planar graph can be 5-colored. PROOF by induction • Recall Sec. 1.4 ex. 16 – Every planar graph has a vertex degree  5. • Consider only connected graphs • Assume all graphs with n-1 vertices (n2) can be 5-colored. • G has a vertex x of degree at most 5. • Delete x to get a graph with n-1 vertices (which by assumption can be 5-colored). • Then reconnect x to the graph and try to color properly. • If the degree of x4, then we can assign x a color. X X If degree of X = 5 Tucker, Applied Combinatorics, Sec 2.4

  11. Class Problem What is the minimum number of guards needed to watch every wall of this gallery? Minimum number in this case is 3 (blue) Tucker, Applied Combinatorics, Sec 2.4

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