5 color theorem
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Five Color Theorem
By Christie Watters and Rita Inamdar
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+The Four Color Problem
It is possible to color any map with only four colors so that notwo adjacent regions will be the same color.
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+History
1852 Francis Guthrie colored a map of England with only fourcolors
Augustus De Morgan a math professor at the UniversityCollege London began to study the problem.
1878 Arthur Cayley brought the question before the LondonMathematical Society.
1879 Alfred Bray Kempe came up with a proof of this theoremwhich was published in theAmerican Journal of Mathematics.
1890 Percy John Heawood while studying at Oxford found acouter example to Kempes proof but not the 4-color
theorem.
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+History Cont.
Throughout the 20
th
century mathematicians working on whatwas now called the four color conjecture for cubic maps.
Now mathematicians attempted to solve the four colorconjecture by finding an an avoidable set S of reducible
configurations.
1976 Kenneth Appel and Wolfgang Haken wrote a computerprogram that constructed 1936 reducible configurations, thus
proving the four color problem!
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+Map and a corresponding
plane graph
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+Kempe Chains
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+Heawood Map
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+Heawood Map
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+Heawood Map
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+5 Color Theorem
Every Planar graph is 5-colorable.
Proof: Let n be the order of the planar graph G. It is obvious thatthe theorem is true for 1 n 5. We will prove this theorem by
induction on n.
Lets Assume every planar graph of order n-1 is 5-colorable, wheren 6. We will show that G is 5-colorable.
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+5 Color Theorem
Corollary 6.6: Every planar graph contains a vertex ofdegree 5 or less.
Therefore G contains a vertex, v such that the deg v 5. G-v is a planar graph of order n-1,therefore our induction
hypothesis states that G-v is 5-colorable.
Let there be a 5-coloring of G-v, where the colors used aredenoted by 1,2,3,4,5.
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5 Color TheoremIf one of these colors in not used to color the neighbors of v, it can beused to color v. This produces a 5 coloring of G. So lets assume all 5
colors are used to color the neighbors of G.
v v
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+5 Color Theorem
Let there be a planar embedding of G and suppose thatv1,v2,v3,v4,v5 are the neighbors of v arranged cyclically about
v where vi has been assigned the coloriwith 1 i 5.
v1
v5
v3
v4
v2
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Let H be a subgraph of G-v induced by the set of verticescolored 1 or 3. Therefore v1,v3V(H). If v1 and v3 belong to
different components of H then we can interchange the colors
of the vertices in H1 containing v1. This produces a 5-coloringof G by assigning color 1 to v.
H:v3
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But wait! Suppose that v1 and v3 are in the same component.(There is a v1-v3 path, P, in G-v.)
The path P and the path (v1
, v, v3
) create a cycle in G.
This will cause either v2 or both v4 and v5 to be enclosed. Therefore there is no v2-v4 path in G-v.
v1
v2
v5v4
v3
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Let F be the subgraph of G-v induced by the set of verticescolored 2 or 4 and let F2 be the component of F containing v2.
Therefore v4
V(F2
), and by interchanging the colors of the
vertices of F2, a 5-coloring of G can be produced by
assigning the color 2 to v.
F:
v2
v4v5 v4
v3
v2v1
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+Bibliography
Chartrand, Gary, Linda Lesniak, and Ping Zhang. Graphs &Digraphs. Boca Raton, FL: CRC, 2011. Print.
Kainen, Paul C. "A Generalization of the 5-color Theorem."Proceedings of the American Mathematical Society. Web. 15 Apr.2012. .
"Nature of Mathematics."Nature of Mathematics. Web. 18 Apr.2012. .