Transcript Section 2.4

Tucker, Applied Combinatorics, Sec. 2.4, Prepared by Whitney and Cody
Section 2.4
Coloring Theorems
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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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Theorem 1:
The vertices in a triangulation of a polygon can be
3-colored.
PROOF: By induction
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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.
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The Art Gallery Problem
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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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Proof:
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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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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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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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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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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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Theorem 5:
It has already been proven that all planar graphs can be 4colored 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
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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.
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If degree of X = 5
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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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