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

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Section 2 4

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

Section 2.4

Coloring Theorems

Tucker, Applied Combinatorics, Sec 2.4


Definitions

Triangulation of a polygon and Cody: 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


Theorem 1

E and Cody

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


The art gallery problem
The Art Gallery Problem and Cody

  • 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


Proof
Proof: and Cody

  • 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


Theorem 2 brook s theorem

Theorem 2 Brook’s Theorem: and Cody

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


Theorem 3

Theorem 3: and Cody

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


  • 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


Theorem 4 vizing s theorem

Theorem 4: Vizing’s Theorem edges with a common end vertex get different colors.

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


Theorem 5

X edges with a common end vertex get different colors.

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


Class problem
Class Problem edges with a common end vertex get different colors.

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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