Graph ColoringStephen Grady, David Eberius, and John Duggan

Three Types of Coloring● Vertex coloring● Edge coloring● Total coloring

Vertex Coloring● Proper vertex coloring - Labeling of all vertices

in a graph such that no two vertices sharing an edge share the same color

Edge Coloring● Proper coloring of edges - labeling of edges

such that no two edges that share the same vertex share the same color.

Total Coloring● Proper total coloring - Coloring of a graph such

that no adjacent vertices or adjacent edges share the same color.

Chromatic Number● Chromatic Number - minimum number of

colors needed to properly color a graph

Brooks’ Theorem● A connected, simple graph’s chromatic

number is no larger than the maximum vertex degree[1]

● Exception: complete and odd cycle graphs[1]

Chromatic Polynomial● Used to count the possible number of ways of

properly vertex coloring a graph.● Example:

P3 can be colored in 12 ways using 3 colors

Chromatic Equivalence Two graphs are chromatically equivalent if they share the same the same chromatic polynomial

History● First presented as a map coloring problem● The problem: Given a planar graph, what is the

minimum number of colors that one can use to color all vertices?

History: 4 vs 5 Coloring● In 1852, Francis Guthrie postulated the four

color conjecture when coloring the counties of England.[2]

● Interestingly, this theorem is not used by map makers.[3]

History: 4 vs 5 ColoringAlfred Kempe followed this up in 1879 by publishing a paper claiming to prove the four color theorem.[2]

11 years later Percy John Heawood showed Kempe was mistaken in his proof but that four colors suffice still held true.[2]

History: Kempe’s Flaw● Where did Kempe’s 4 color theorem go

wrong?

History: 4 vs 5 ColoringThe four color theorem was finally proven in 1976 by Kenneth Appel and Wolfgang Haken[4,5]. (It should be said that this was proof was aided by a computer, so there was some controversy surrounding this proof.)

Complexity● Decision: NP-complete● Optimization: NP-hard

Algorithms ● Greedy Algorithms● Brute force algorithm● Parallel Processing algorithms

Greedy Coloring● Given a vertex order, assign each vertex the

first available color● Fast, but potentially poor

Vertex Ordering● Any graph has a vertex ordering for greedy

coloring that yields an optimal coloring● Finding this ordering for arbitrary graphs is

NP-hard

Welsh-Powell Algorithm● Provides an upper-bound on the number of

colors used by greedy coloring[6]

● Order vertices by descending degree

Welsh-Powell Example

Welsh-Powell Example

Chordal Graphs● All cycles of 4 or more vertices have a chord● Chord - an edge not in the cycle which

connects 2 of the vertices

Perfect Elimination Ordering

● A graph is chordal iff it has a perfect elimination ordering[7]

● Each vertex and its neighbors which follow it in the order form a clique[7]

● This ordering can be found using lexicographic BFS[7]

Lexicographic BFS1. Begin with all vertices in a single set2. Remove a vertex v from the first set3. Place this vertex at the end of the order4. Divide each set into two subsets: neighbors of v and

non-neighbors of v.5. Repeat 2-4 until all vertices are in the order

Note: if we reverse this ordering, we get a perfect elimination ordering (if one exists)

Brute ForceThe brute force search is highly inefficient as it runs through each possibility of kn where k is the number of colors and n the number of vertices.

Brute Force● The key factor for the inefficiency of the

brute force algorithm is the sheer number of colorings to check.

● Is there a way to reduce the number of colorings?

Brute Force● Number of Colorings Used:

○ C = 1 -> 1 coloring○ 1 < C < V -> ???○ C = V -> 1 coloring

● This is still a huge number of colorings, so it’s still extremely slow!

Brute Force Algorithm● Generate all possible combinations of

‘guarantee’ spots for all C where 1 < C < V○ For each of these combinations, generate all

possible permutations of C colors from 0 to C-1■ For each of the above permutations, generate all

possible combinations of colors in the remaining V-C spots by counting in base C with V-C digits.

Brute Force Algorithm● Test each coloring combination for validity.● If a valid coloring is found, stop generating

possibilities, because the current C is the chromatic number.○ If this is true, what order did I test my colorings in?

Parallel Processing● Performing work on multiple processors at

the same time.● You must divide tasks up such that there

aren’t any dependences between tasks that are running at the same time.○ DMV Analogy○ What are dependences?

Matrix Matrix Multiply● Multiply two matrices

together.● This algorithm is one

of the quintessential parallel programming algorithms.

Matrix Matrix Multiply● The Algorithm:

○ For each row of the left side matrix, multiply the ith element with the ith element of the corresponding column in the right matrix.

○ The sum of these multiplications is put at the [r][c] spot of the result matrix.

for(int r = 0; r < dim; r++) { for(int c = 0; c < dim; c++) { sum = 0; for(int i = 0; i < dim; i++) { sum += matrix[r][i] * matrix[i][c]; } result[r][c] = sum; } }

Parallel Processing Algorithms

● Times are in seconds.

● Sizes are of M by M matrices.

● 4 Hyperthreaded cores were used.

Parallel Brute Force● Used nearly the same algorithm as the serial

version, with the following differences:○ Generate N choices for the ‘guarantee’ spots, where

N is the number of parallel threads.○ Assign each thread to generate the subsequent

possible colorings for these spots and test whether they are valid colorings.

○ Merge all threads and check for completion.

Implementations● Greedy coloring

○ Random vertex ordering○ Welsh-Powell ordering○ Ordering by lexicographic BFS

● Brute force● Parallelized brute force

Results (greedy)

Results (greedy)Random Welsh-Powell Lex. BFS

random_5_5.txt 3 3 3

random_10_20.txt 4 4 4

random_20_35.txt 4 4 5

random_40_100.txt 5 4 5

random_100_500.txt 6 6 7

random_200_600.txt 6 5 6

random_500_1500.txt 6 6 6

Results (brute force)● Small graphs run quickly,

but the run times grow exponentially.○ Note: Timings are in

seconds.○ The longest runs for each

category took 137.68 and 68.87 minutes respectively.

Results (parallel)● Timings are in

seconds.● All graphs were

cliques.● Speedup less than 1

means parallel is slower.

Applications● Sudoku● Register Allocation● Scheduling Problem

SudokuA game in which the goal is to create a 9x9 chart where no number shares a column, row or 3x3 square.

Sudoku

Sudoku● Sudoku can be represented as a graph coloring

problem● Transform the board into a graph with 81

vertices where two vertices that shares a column, row or 3x3 square are connected by an edge. Now color using 9 colors.

Sudoku

Register Allocation● At compile time, the compiler allocates registers

to certain variables in a program. Multiple variables can exist in the same register but they cannot be called at the same time, so allocating them becomes a coloring problem.

Register Allocation

Scheduling ProblemGraph coloring can be used to find the minimum number of time slots needed to create a schedule with no time conflicts

Example● Finals will be here soon so we’ll use that as our

example.● For each Student list out what classes are being

taken● Student1 {biology, diff. eq, bio stats, graph algo }● Student2 {English lit, journalism, biology, pchem1}● Student3 {bio stats, modern physics, modern physics

lab}

Example continued● Each vertex represents a class and an edge

between vertices represents a student taking both.

So our example becomes….

Example continued

Example Continued● Now we simply color to determine how many

time slots are required to make sure there are now time conflicts.

The above graph becomes...

Final ScheduleFor the example there should be 4 time slots.

Open Problems● Is there a polynomial time algorithm for

checking if a k-coloring for a graph exists?

Open Problems● Erdős-Faber-Lovász Conjecture - the union of n

pairwise edge-disjoint complete graphs with n vertices is n-colorable

Open Problems● Reed’s upper bound[8]:

(G) ≤ (1 + (G) + (G)) / 2

References1. Brooks, R.L. On Colouring the Nodes of a Network. Mathematical Proceedings of the

Cambridge Philosophical Society, April 1941.2. Thomas, Robin. The Four Color Theorem. School of Mathematics, Georgia Institute of

Technology. http://people.math.gatech.edu/~thomas/FC/fourcolor.html.3. Wilson, Robin. Four Colors Suffice. London: Penguin Books, 2002.4. Appel, Kenneth and Wolfgang Haken. Every Planar Map is Four Colorable Part I. Discharging.

Illinois Journal of Mathematics, 1977.5. Appel, Kenneth et al. Every Planar Map is Four Colorable Part II. Reducibility. Illinois Journal

of Mathematics, 1977.6. Welsh, D.J.A. and M.B. Powell. An Upper Bound for the Chromatic Number of a Graph and Its

Application to Timetabling Problems. The Computer Journal, 1967.7. Rose, Donald J. et al. Algorithmic Aspects of Vertex Elimination on Graphs. SIAM Journal on

Computing, 1976.8. http://www.math.illinois.edu/~dwest/openp/

Email answers to sgrady3@vols.utk.edu.

1. Use the lexicographical BFS algorithm to produce an ordering for the following graph (details on next page):

Homework

0

2

13

5

4 6

HomeworkFor question 1, begin with vertex 6. When breaking ties, select the lowest-numbered vertex. See http://en.wikipedia.org/wiki/Lexicographic_breadth-first_search for more details on the algorithm itself.

Homework2. What is the chromatic number of this graph?

A

B

C

D

E

F

G

H

I

Homework3. Given 9 classes and 3 students with schedules:

Student 1 {1,2,6,9}Student 2 {2,3,4,7}Student 3 {1,5,8,9}

How many time slots are needed to guarantee no schedule conflicts?