G r a p h C o l o r i n g
CS 594
Stephen Grady

-
Terminology
-History
4 Color Theorem
Kempe’s flawed proof
-Simple Bounds on chromatic number
-Coloring on Digraphs
-Applications

-Color: Any labeling assigned to an element of a graph.
-Proper coloring: No color may be adjacent to itself.

-Chromatic number ( χ(G)): Minimum colors needed to properly color graph.
-Chromatic Equivalency: Two graphs are chromatically equivalent if they have the same χ(G)

-The possible number of ways to color a graph using k colors.
Example K
3+1 colors can be colored 12 different ways with 3

-Vertex-Vertices are labeled
-Edge-Edges are labeled
-Total-Both vertices and edges are labeled
-Many more...

-Coloring most concerned with
-Other colorings can be transformed to vertex coloring
Edge: color vertices of line graph
Planer: color vertices of dual

-First studied as a map coloring problem.
-What is the minimum number of colors needed to color a map. Or stated another way, what is the minimum number of colors needed to color a planar graph?

-Francis Guthrie first noticed he could color the map of the counties of England using only 4 colors.
-Naturally the question of if this holds true for all maps (planar graphs) arose.

-The question was passed along in 1852
Guthrie's brother->Augustus de Morgan->William
Hamilton
-Eventually it was brought up to the London
Mathematical Society in 1879
-That same year Alfred Kempe published a paper claiming to contain a proof that 4 colors suffice.

-1849-1922
-Trinity College, Cambridge
-22 nd wrangler
-1877: Flawed “straight line linkage” proof, though ideas basis of proof in 2002
-1879: Flawed 4 color theorem proof though ideas basis of proof in 1976

-1861-1955
-Exeter College, Oxford
-Much of life dedicated to
4 color theorem
-1890:Disproved Kempe's
4 color theorem proof
-1890:Established 5 color theorem based on Kempe's work

-Kempe's proof rested on the properties of what are known as Kempe chains.
-A Kempe chain is a bicolored path between any two non-adjacent vertices.

-Assume Δ(G) ≤5 (There is a proof but I’ll leave that to whomever covers planarity)
-Consider the smallest planar graph G in terms of
|V| that requires 5 colors to color properly.
-Delete any vertex v in G creating G'. Since G' is smaller than G, G' is 4 colorable.
-Color all vertices in G' using 4 colors.

-Now add back v to G' recreating G.
-There are three cases to consider when deciding how to color v with one of the four colors used.
deg (v) =1,2 or 3 deg (v)=4 deg (v)=5

v has degree 1,2 or 3.
-Since the neighbors of v only take up to 3 of the possible colors in this case coloring v is trivial.

-v has degree 4 and neighbors a, b, c and d .
-If the neighbors of v do not use all 4 colors, then coloring v is trivial.
-If all 4 colors are used by the neighbors of v then choose any two neighbors a ,c of v and consider the subgraph H of G induced by the colors of a and c .
-This creates two possibilities

-There does not exist a Kempe chain between a and c
-If this is the case then consider the subgraph J of
H induced by taking all vertices that have a path to a .
-Perform a color swap on J and now a and c share the same color and v may be colored with the remaining color.

-There does exist a Kempe chain between a and c .
-Now consider the subgraph H' of G induced by the colors of b and d.
-Because G is planar there cannot exist a Kempe chain from b to d.
-Simply perform a color swap on b as in subcase i.

-v has degree 5 with neighbors a,b,c,d and e .
-Add edges to all neighbors of v while staying planar
-Assume all 4 colors are used by neighbors of v .
-Just like in case 2 there exist 2 subcases.

-Consider three neighbors b, e and d.
-As in case 2 consider the subgraph H of G induced by the colors of b and e .
-If no Kempe chain, do a color swap.
-If Kempe chain, repeat for b and d.

-Both b to d and b to e have a kempe chain.
-Consider the subgraph H' of G induced by colors on a and d. As with case 2 because G is planar there cannot exist a Kempe chain from a to d so a color swap can be performed on a.
Repeat for c and e .

What if the Kempe chain's b to d and b to e cross?

-Finally proven in 1976 by Kenneth Appel and
Wolfgang Haken at University of Illinois.
-First proof using a computer as an aid.

-Used the idea of an unavoidable set of reducible configurations
-Had to check 1,936 graphs to prove minimum counterexample to 4 color theorem could not exist.

-Decision: Given a graph G can it be colored with k colors? NP-Complete
-Optimization: What is χ(G)? NP-Hard

-Brooke's Theorem
-Clique number

χ(G) ≤ Δ(G)
Except for Kn and C2n+1
χ(G) ≤ Δ(G)+1

χ(G) >= ω(G)
-A clique of size Kn must be colored with n colors.

-There exists a triangle free graph with arbitrarily high χ(G).
-Generalized with Mycielski graphs.

-Gallai-Roy Theorem
χ(G) =L+1 where L is the longest path in its shortest orientation.

-Applications are generally those that must satisfy some constraint
Scheduling
Register allocation
Determining if graph is bipartite
Sudoku


Used to find minimum number of time slots needed with no time conflicts.

Each time slot represented by a color.

Each edge represents time conflict


In an attempt to optimize code, compilers will allocate multiple variables to the same register.

However, multiple variables allocated to the same register cannot be called at the same time.

Naturally this becomes a coloring problem.


Bipartite Graphs always have χ(G)=2
 Can check if graph is 2 colorable in linear time


Why study Sudoku

Methods for solving Sudoku can be generalized to solve problems like protein folding.

 Erdos-Faber-Lovasz Conjecture
 Reed’s upper bound

Can n kn graphs each sharing only one vertex be colored with n colors.

χ(G) ≤ (1+Δ(G)+ω(G))/2

-Prove that for any planar graph 5 colors suffice
(Assume Δ(G) ≤5 )
-Which two of these graphs are chromatically equivalent?
k
4,4
, P
7
, k
5
, Peterson graph
-How many ways can a k6 be colored using 7 colors?
Email: sgrady3@vols.utk.edu

1.
https://en.wikipedia.org/wiki/Graph_coloring
2.
https://en.wikipedia.org/wiki/Alfred_Kempe
3.
https://en.wikipedia.org/wiki/Percy_John_Heawood
4.
https://en.wikipedia.org/wiki/Wrangler_%28University_of_Ca mbridge%29
5.
https://en.wikipedia.org/wiki/Four_color_theorem
6.
https://en.wikipedia.org/wiki/Mycielskian
7.
https://en.wikipedia.org/wiki/Gallai%E2%80%93Hasse%E2%80
%93Roy%E2%80%93Vitaver_theorem
8.
http://www.math.illinois.edu/~dwest/openp/
9.
http://math.ucsb.edu/~padraic/ucsb_2014_15/math_honors
_f2014/math_honors_f2014_lecture4.pdf
10.
https://en.wikipedia.org/wiki/Sudoku

11. https://en.wikipedia.org/wiki/Bipartite_graph
12. http://www.math.rutgers.edu/~sk1233/courses/grap htheory-F11/planar.pdf
13.
http://www.skidmore.edu/~adean/MC3021309/Print
Slides/MC302_131203_P.pdf
14. Ercsey-Ravasz, M. and Z. Toroczkai (2012). "The Chaos
Within Sudoku." Scientific Reports 2: 725.