Graph Coloring CS 594 Stephen Grady

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

Terminology graph. itself.

Terminology needed to properly color graph. chromatically equivalent if they have the same χ(G)

Chromatic Polynomial using k colors. Example K3+1 can be colored 12 different ways with 3 colors

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

Vertex coloring -Coloring most concerned with coloring Edge: color vertices of line graph Planer: color vertices of dual

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

4 Color Theorem colors. maps (planar graphs) arose.

4 Color Theorem -The question was passed along in 1852 Guthrie's brother->Augustus de Morgan->William Hamilton Mathematical Society in 1879 claiming to contain a proof that 4 colors suffice.

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

Percy John Heawood -1861-1955 -Exeter College, Oxford 4 color theorem 4 color theorem proof based on Kempe's work

Kemp's Flawed 4 Color Theorem -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.

Kempe's Argument that to whomever covers planarity) |V| that requires 5 colors to color properly. smaller than G, G' is 4 colorable. -Color all vertices in G' using 4 colors.

Kempe's Argument -Now add back v to G' recreating G. how to color v with one of the four colors used. deg(v)=1,2 or 3 deg(v)=4 deg(v)=5

Case 1 -v has degree 1,2 or 3. possible colors in this case coloring v is trivial.

Case 2 -v has degree 4 and neighbors a, b, c and d. coloring v is trivial. and c. -This creates two possibilities

Case 2 Subcase i and c to a. remaining color.

Case 2 Subcase ii c. the colors of b and d. chain from b to d.

Case 3 -v has degree 5 with neighbors a,b,c,d and e. planar -Assume all 4 colors are used by neighbors of v. -Just like in case 2 there exist 2 subcases.

Case 3 Subcase i -Consider three neighbors b, e and d. induced by the colors of b and e. -If no Kempe chain, do a color swap. -If Kempe chain, repeat for b and d.

Case 3 Subcase ii -Both b to d and b to e have a kempe chain. a color swap can be performed on a. -Repeat for c and e.

So, why is it flawed?

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

4 Color Theorem Wolfgang Haken at University of Illinois. -First proof using a computer as an aid.

4 Color Theorem -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.

Complexity k colors? NP-Complete -Optimization: What is χ(G)? NP-Hard

Bounds on χ(G) -Brooke's Theorem -Clique number

Brooke's Theorem χ(G) ≤ Δ(G) Except for Kn and C2n+1 χ(G) ≤ Δ(G)+1

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

Mycielski's Theorem high χ(G). -Generalized with Mycielski graphs.

Coloring on Digraphs -Gallai-Roy Theorem shortest orientation.

Applications must satisfy some constraint Scheduling Register allocation Determining if graph is bipartite Sudoku

Scheduling 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

Register Allocation 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.

Register Allocation

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

Sudoku

Sudoku Graph Transformation

Coloring a Sudoku Graph

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

Open Problems Erdos-Faber-Lovasz Conjecture Reed’s upper bound

Erdos-Faber-Lovasz Conjecture Can n kn graphs each sharing only one vertex be colored with n colors.

Reed’s Upper Bound χ(G) ≤ (1+Δ(G)+ω(G))/2

Homework -Prove that for any planar graph 5 colors suffice (Assume Δ(G) ≤5) -Which two of these graphs are chromatically equivalent? k4,4, P7, k5, Peterson graph -How many ways can a k6 be colored using 7 colors? Email: sgrady3@vols.utk.edu

References https://en.wikipedia.org/wiki/Graph_coloring https://en.wikipedia.org/wiki/Alfred_Kempe https://en.wikipedia.org/wiki/Percy_John_Heawood https://en.wikipedia.org/wiki/Wrangler_%28University_of_Cambridge%29 https://en.wikipedia.org/wiki/Four_color_theorem https://en.wikipedia.org/wiki/Mycielskian https://en.wikipedia.org/wiki/Gallai%E2%80%93Hasse%E2%80%93Roy%E2%80%93Vitaver_theorem http://www.math.illinois.edu/~dwest/openp/ http://math.ucsb.edu/~padraic/ucsb_2014_15/math_honors_f2014/math_honors_f2014_lecture4.pdf https://en.wikipedia.org/wiki/Sudoku

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