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A Variation on the Four Color Problem
Jennifer Li
Department of MathematicsLouisiana State University
Baton Rouge
History
(1852) Francis Guthrie:
“Do four colors suffice to color every planar map so that adjacentcountries receive different colors?”
The Four Color Problem or The Map Coloring Problem.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 2 / 24
History
(1852) Francis Guthrie:
“Do four colors suffice to color every planar map so that adjacentcountries receive different colors?”
The Four Color Problem or The Map Coloring Problem.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 2 / 24
History
(1852) Francis Guthrie:
“Do four colors suffice to color every planar map so that adjacentcountries receive different colors?”
The Four Color Problem or The Map Coloring Problem.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 2 / 24
Planar Maps
Examples:
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 3 / 24
Example
The planar map below is colored with four colors.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 4 / 24
From Map to Graph
We can depict maps abstractly as graphs.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 5 / 24
From Map to Graph
We can depict maps abstractly as graphs.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 5 / 24
From Map to Graph
We can depict maps abstractly as graphs.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 5 / 24
Chromatic Number
Least number of colors needed, denoted by χ(G )
Examples:
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 6 / 24
Chromatic Number
Least number of colors needed, denoted by χ(G )Examples:
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 6 / 24
Chromatic Number
Least number of colors needed, denoted by χ(G )Examples:
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 6 / 24
Another Coloring Problem
Many attempts to prove the Four Color Problem!
1879 A. Kempe1890 P. Heawood
What Heawood did:
1.
2.
3. Another question:
How many colors are required for graphs embedded on surfaces other thanthe plane?
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 7 / 24
Another Coloring Problem
Many attempts to prove the Four Color Problem!
1879 A. Kempe
1890 P. Heawood
What Heawood did:
1.
2.
3. Another question:
How many colors are required for graphs embedded on surfaces other thanthe plane?
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 7 / 24
Another Coloring Problem
Many attempts to prove the Four Color Problem!
1879 A. Kempe1890 P. Heawood
What Heawood did:
1.
2.
3. Another question:
How many colors are required for graphs embedded on surfaces other thanthe plane?
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 7 / 24
Another Coloring Problem
Many attempts to prove the Four Color Problem!
1879 A. Kempe1890 P. Heawood
What Heawood did:
1.
2.
3. Another question:
How many colors are required for graphs embedded on surfaces other thanthe plane?
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 7 / 24
Another Coloring Problem
Many attempts to prove the Four Color Problem!
1879 A. Kempe1890 P. Heawood
What Heawood did:
1.
2.
3. Another question:
How many colors are required for graphs embedded on surfaces other thanthe plane?
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 7 / 24
Another Coloring Problem
Many attempts to prove the Four Color Problem!
1879 A. Kempe1890 P. Heawood
What Heawood did:
1.
2.
3. Another question:
How many colors are required for graphs embedded on surfaces other thanthe plane?
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 7 / 24
Another Coloring Problem
Many attempts to prove the Four Color Problem!
1879 A. Kempe1890 P. Heawood
What Heawood did:
1.
2.
3. Another question:
How many colors are required for graphs embedded on surfaces other thanthe plane?
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 7 / 24
Another Coloring Problem
Many attempts to prove the Four Color Problem!
1879 A. Kempe1890 P. Heawood
What Heawood did:
1.
2.
3. Another question:
How many colors are required for graphs embedded on surfaces other thanthe plane?
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 7 / 24
Other Surfaces
Sphere = simplest surface
Sphere + n handles = new surface
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 8 / 24
Other Surfaces
Sphere = simplest surface
Sphere + n handles = new surface
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 8 / 24
Other Surfaces (continued)
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 9 / 24
Other Surfaces (continued)
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 10 / 24
Embedding Graphs on Other Surfaces
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 11 / 24
Embedding Graphs on Other Surfaces
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 11 / 24
Embedding Graphs on Other Surfaces
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 11 / 24
Embedding Graphs on Other Surfaces (continued)
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 12 / 24
Embedding Graphs on Other Surfaces (continued)
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 12 / 24
Stereographic Projection
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 13 / 24
Graph Embedding Examples
K5 is not planar, but is toroidal:
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 14 / 24
Graph Embedding Examples
K5 is not planar, but is toroidal:
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 14 / 24
Graph Embedding Examples
K5 is not planar, but is toroidal:
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 14 / 24
Faces of a Graph
Example:
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 15 / 24
Faces of a Graph
Example:
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 15 / 24
Euler-Poincare Formula
v = verticese = edgesf = faces
For a connected, nonempty planar graph,
v − e + f = 2.
The Euler genus g is defined to be
g = 2− (v − e + f )
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 16 / 24
Euler-Poincare Formula
v = verticese = edgesf = faces
For a connected, nonempty planar graph,
v − e + f = 2.
The Euler genus g is defined to be
g = 2− (v − e + f )
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 16 / 24
Euler-Poincare Formula
v = verticese = edgesf = faces
For a connected, nonempty planar graph,
v − e + f = 2.
The Euler genus g is defined to be
g = 2− (v − e + f )
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 16 / 24
Example
plane: g = 2− (v − e + f ) = 2− 2 = 0
Torus: g = 2− (5− 10 + 5) = 2− 0 = 2Klein bottle: g = 2− 0 = 2Projective plane: g = 2− 1 = 1
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 17 / 24
Example
plane: g = 2− (v − e + f ) = 2− 2 = 0
Torus: g = 2− (5− 10 + 5) = 2− 0 = 2
Klein bottle: g = 2− 0 = 2Projective plane: g = 2− 1 = 1
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 17 / 24
Example
plane: g = 2− (v − e + f ) = 2− 2 = 0
Torus: g = 2− (5− 10 + 5) = 2− 0 = 2Klein bottle: g = 2− 0 = 2
Projective plane: g = 2− 1 = 1
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 17 / 24
Example
plane: g = 2− (v − e + f ) = 2− 2 = 0
Torus: g = 2− (5− 10 + 5) = 2− 0 = 2Klein bottle: g = 2− 0 = 2Projective plane: g = 2− 1 = 1
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 17 / 24
Heawood’s Conjecture
Heawood used the Euler-Poincare Formula to show:
If G is a loopless graph embedded in surface S with Euler genus g , then
χ(G ) ≤⌊(
7+√24g+12
)⌋
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 18 / 24
Heawood’s Conjecture
Heawood used the Euler-Poincare Formula to show:If G is a loopless graph embedded in surface S with Euler genus g , then
χ(G ) ≤⌊(
7+√24g+12
)⌋
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 18 / 24
Heawood’s Conjecture
Heawood used the Euler-Poincare Formula to show:If G is a loopless graph embedded in surface S with Euler genus g , then
χ(G ) ≤⌊(
7+√24g+12
)⌋
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 18 / 24
Example
Let S be the torus, which has Euler genus g = 2.
By Heawood’s Formula: For any loopless graph G embedded in S , we have
χ(G ) ≤ 12(7 +
√24 · 2 + 1) = 1
2(7 + 7) = 7
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 19 / 24
Proof of Conjecture
Heawood proved the cases where g > 0.
For g = 0 (sphere):
χ(G ) ≤ 12(7 +
√24 · 0 + 1) = 1
2(7 + 1) = 4
The Four Color Problem!
BUT...Heawood could not prove this case.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 20 / 24
Proof of Conjecture
Heawood proved the cases where g > 0.
For g = 0 (sphere):
χ(G ) ≤ 12(7 +
√24 · 0 + 1) = 1
2(7 + 1) = 4
The Four Color Problem!
BUT...Heawood could not prove this case.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 20 / 24
Proof of Conjecture
Heawood proved the cases where g > 0.
For g = 0 (sphere):
χ(G ) ≤ 12(7 +
√24 · 0 + 1) = 1
2(7 + 1) = 4
The Four Color Problem!
BUT...Heawood could not prove this case.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 20 / 24
Proof of Conjecture
Heawood proved the cases where g > 0.
For g = 0 (sphere):
χ(G ) ≤ 12(7 +
√24 · 0 + 1) = 1
2(7 + 1) = 4
The Four Color Problem!
BUT...Heawood could not prove this case.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 20 / 24
Proof of Conjecture
Heawood proved the cases where g > 0.
For g = 0 (sphere):
χ(G ) ≤ 12(7 +
√24 · 0 + 1) = 1
2(7 + 1) = 4
The Four Color Problem!
BUT...
Heawood could not prove this case.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 20 / 24
Proof of Conjecture
Heawood proved the cases where g > 0.
For g = 0 (sphere):
χ(G ) ≤ 12(7 +
√24 · 0 + 1) = 1
2(7 + 1) = 4
The Four Color Problem!
BUT...Heawood could not prove this case.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 20 / 24
Best Bound?
Every map on surface S with Euler genus g can be colored with at most⌊(7+
√24g+12
)⌋colors.
Is this the best bound? Yes, for every surface EXCEPT Klein bottle!
The formula gives χ(G ) ≤ (12)(7 +√
24 · 2 + 1) = (12)(7 + 7) = 7
1934, P. Franklin: Klein bottle requires at most 6 colors.
1954, G. Ringel: Heawood conjecture gives the best bound for every othersurface.1967, T. Youngs: Simplified Ringel’s solution.1974, Ringel-Youngs Theorem published.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 21 / 24
Best Bound?
Every map on surface S with Euler genus g can be colored with at most⌊(7+
√24g+12
)⌋colors.
Is this the best bound?
Yes, for every surface EXCEPT Klein bottle!
The formula gives χ(G ) ≤ (12)(7 +√
24 · 2 + 1) = (12)(7 + 7) = 7
1934, P. Franklin: Klein bottle requires at most 6 colors.
1954, G. Ringel: Heawood conjecture gives the best bound for every othersurface.1967, T. Youngs: Simplified Ringel’s solution.1974, Ringel-Youngs Theorem published.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 21 / 24
Best Bound?
Every map on surface S with Euler genus g can be colored with at most⌊(7+
√24g+12
)⌋colors.
Is this the best bound? Yes, for every surface EXCEPT
Klein bottle!
The formula gives χ(G ) ≤ (12)(7 +√
24 · 2 + 1) = (12)(7 + 7) = 7
1934, P. Franklin: Klein bottle requires at most 6 colors.
1954, G. Ringel: Heawood conjecture gives the best bound for every othersurface.1967, T. Youngs: Simplified Ringel’s solution.1974, Ringel-Youngs Theorem published.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 21 / 24
Best Bound?
Every map on surface S with Euler genus g can be colored with at most⌊(7+
√24g+12
)⌋colors.
Is this the best bound? Yes, for every surface EXCEPT Klein bottle!
The formula gives χ(G ) ≤ (12)(7 +√
24 · 2 + 1) = (12)(7 + 7) = 7
1934, P. Franklin: Klein bottle requires at most 6 colors.
1954, G. Ringel: Heawood conjecture gives the best bound for every othersurface.1967, T. Youngs: Simplified Ringel’s solution.1974, Ringel-Youngs Theorem published.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 21 / 24
Best Bound?
Every map on surface S with Euler genus g can be colored with at most⌊(7+
√24g+12
)⌋colors.
Is this the best bound? Yes, for every surface EXCEPT Klein bottle!
The formula gives χ(G ) ≤ (12)(7 +√
24 · 2 + 1) = (12)(7 + 7) = 7
1934, P. Franklin: Klein bottle requires at most 6 colors.
1954, G. Ringel: Heawood conjecture gives the best bound for every othersurface.1967, T. Youngs: Simplified Ringel’s solution.1974, Ringel-Youngs Theorem published.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 21 / 24
Best Bound?
Every map on surface S with Euler genus g can be colored with at most⌊(7+
√24g+12
)⌋colors.
Is this the best bound? Yes, for every surface EXCEPT Klein bottle!
The formula gives χ(G ) ≤ (12)(7 +√
24 · 2 + 1) = (12)(7 + 7) = 7
1934, P. Franklin: Klein bottle requires at most 6 colors.
1954, G. Ringel: Heawood conjecture gives the best bound for every othersurface.1967, T. Youngs: Simplified Ringel’s solution.1974, Ringel-Youngs Theorem published.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 21 / 24
Best Bound?
Every map on surface S with Euler genus g can be colored with at most⌊(7+
√24g+12
)⌋colors.
Is this the best bound? Yes, for every surface EXCEPT Klein bottle!
The formula gives χ(G ) ≤ (12)(7 +√
24 · 2 + 1) = (12)(7 + 7) = 7
1934, P. Franklin: Klein bottle requires at most 6 colors.
1954, G. Ringel: Heawood conjecture gives the best bound for every othersurface.
1967, T. Youngs: Simplified Ringel’s solution.1974, Ringel-Youngs Theorem published.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 21 / 24
Best Bound?
Every map on surface S with Euler genus g can be colored with at most⌊(7+
√24g+12
)⌋colors.
Is this the best bound? Yes, for every surface EXCEPT Klein bottle!
The formula gives χ(G ) ≤ (12)(7 +√
24 · 2 + 1) = (12)(7 + 7) = 7
1934, P. Franklin: Klein bottle requires at most 6 colors.
1954, G. Ringel: Heawood conjecture gives the best bound for every othersurface.1967, T. Youngs: Simplified Ringel’s solution.
1974, Ringel-Youngs Theorem published.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 21 / 24
Best Bound?
Every map on surface S with Euler genus g can be colored with at most⌊(7+
√24g+12
)⌋colors.
Is this the best bound? Yes, for every surface EXCEPT Klein bottle!
The formula gives χ(G ) ≤ (12)(7 +√
24 · 2 + 1) = (12)(7 + 7) = 7
1934, P. Franklin: Klein bottle requires at most 6 colors.
1954, G. Ringel: Heawood conjecture gives the best bound for every othersurface.1967, T. Youngs: Simplified Ringel’s solution.1974, Ringel-Youngs Theorem published.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 21 / 24
A Correct Solution!
1977, Appel, Haken: The Four Color Problem
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 22 / 24
A Correct Solution!
1977, Appel, Haken: The Four Color Problem
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 22 / 24
A Correct Solution!
1977, Appel, Haken: The Four Color Problem
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 22 / 24
Summary
The Four Color Problem: unsolved for nearly 100 years!
Moreover,Four Color Problem: 1200 hours of computer time on University of
Illinois supercomputer.Heawood’s conjecture: application of the quadratic formula.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 23 / 24
Summary
The Four Color Problem: unsolved for nearly 100 years!
Moreover,
Four Color Problem: 1200 hours of computer time on University ofIllinois supercomputer.
Heawood’s conjecture: application of the quadratic formula.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 23 / 24
Summary
The Four Color Problem: unsolved for nearly 100 years!
Moreover,Four Color Problem:
1200 hours of computer time on University ofIllinois supercomputer.
Heawood’s conjecture: application of the quadratic formula.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 23 / 24
Summary
The Four Color Problem: unsolved for nearly 100 years!
Moreover,Four Color Problem: 1200 hours of computer time on University of
Illinois supercomputer.
Heawood’s conjecture: application of the quadratic formula.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 23 / 24
Summary
The Four Color Problem: unsolved for nearly 100 years!
Moreover,Four Color Problem: 1200 hours of computer time on University of
Illinois supercomputer.Heawood’s conjecture:
application of the quadratic formula.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 23 / 24
Summary
The Four Color Problem: unsolved for nearly 100 years!
Moreover,Four Color Problem: 1200 hours of computer time on University of
Illinois supercomputer.Heawood’s conjecture: application of the quadratic formula.
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 23 / 24
Thanks to:My project advisor, Professor Oporowski!Federico Salmoiraghi!
Jennifer Li (Louisiana State University) A Variation on the Four Color Problem May 2, 2015 24 / 24