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

2. Tucker, Applied Combinatorics, Sec 2.4
Definitions:
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).
Chromatic number: The smallest number of colors that can be used in a coloring of a graph G
Triangulation of G
Chromatic number = 3
Symbols:
Let the symbol (G) denote the chromatic number of the graph G.
Let the symbol r denote the largest integer  r.
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Tucker, Applied Combinatorics, Sec 2.4

3. 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. E E E
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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
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Tucker, Applied Combinatorics, Sec 2.4

5. Tucker, Applied Combinatorics, Sec 2.4
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.
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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.
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Tucker, Applied Combinatorics, Sec 2.4

7. Tucker, Applied Combinatorics, Sec 2.4
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
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Tucker, Applied Combinatorics, Sec 2.4

8. 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…
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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.
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Tucker, Applied Combinatorics, Sec 2.4

10. Tucker, Applied Combinatorics, Sec 2.4
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 X X
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If degree of X = 5
Tucker, Applied Combinatorics, Sec 2.4

11. Tucker, Applied Combinatorics, Sec 2.4
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)
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Tucker, Applied Combinatorics, Sec 2.4