Graph coloring: From maps to wireless networks and job schedulingMagnús M. Halldórsson

University of Iceland

Outline of talk

Map coloring and graph coloring Applications of coloring Distance-2 coloring problem

Planar graphs

The map coloring problem

Color the countries with the fewest number of colors

Different colors on adjacent countries

“Folklore”: 4 colors always suffice

Map Coloring and Graph coloring

(G) = minimum number of colors needed for graph G

A student of mine asked me today to give him a reason for a fact which I did not know was a fact - and do not yet. He says that if a figure be anyhow divided and the compartments differently coloured so that the figures with any portion of the common boundary line are differently coloured - four colours may be wanted, but not more - the following is the case in which four colours are wanted. Query cannot a necessity for five or more be invented.

DeMorgan wrote the following to Hamilton in 1852:

Francis Guthrie noticed that “four colours suffice”.His brother Frederick Guthrie communicated this to DeMorgan:

The early history of the problem 1852 - Guthrie & DeMorgan 1879 - 1880 - first proofs announced

Kempe + Tait 1890 - 1891 - first proofs renounced

Heawood & Petersen 1900’s-1970’s - reduction methods.

Birkhoff, Whitney, Ore, Heesch… “Almost every great mathematician has worked on

it at one time or another” (Birkhoff)

4-coloring problem

“The entire development of the subject of graph theory can be traced back to attempts to solve the 4-color problem.” R. Wilson, Four Colors Suffice

Many misguided attempts

Herman Minkowski, the distinguished number-theorist, was lecturing on topology at Göttingen, and mentioned the 4-color problem:

‘This theorem has not been proved, but that is because only mathematicians of the third rank have occupied themselves with it. I believe that I can prove it.’

And began to work out his demonstration on the spot. After several weeks, he announced:

‘Heaven is angered by my arrogance’.

Proof of the 4-Color Theorem[Appel, Haken, 1976]

Reduce to 1476 configurations of regions (using Heesch’s concepts of “reducibility” and “discharging”).

More than 1200 hours of computer time on an IBM 360.

The first proof of a theorem using a computer. Cannot be verified by other mathematicians, but

is now generally accepted. Another, more verifiable proof in 1994

Outline of talk

Map coloring and graph coloring Applications of coloring Distance-2 coloring problem

Planar graphs

Is Graph Coloring Useful?

The original motivation was suspect:A sample of atlases in the large collection of the Library of Congress indicates no tendency to minimize the number of colors used. Maps utilizing only four colours are rare, and those that do usually require only three. Books on cartography and the history of map-making do not mention the four-color property…

Kenneth May, math historian, 1956

On Utility

Magnus’s Axiom of Importance: If a problem has a clean, compact, and elegant

definition, then it is important, and applications will show up

Corollary: The longer it takes to explain, the less interesting it is (at least theoretically speaking)

Euler’s formula

17 different proofs http://www.ics.uci.edu/~eppstein/junkyard/euler/

e 3v-6 Each face has at least 3 edges,

& each edge occurs in 2 faces Edge-face incidences: 3f 2e

Average degree = 2e/v 6 – 12/v < 6. Ergo, some vertex has degree 5

v-e+f = 2

#Vertices - #Edges + #faces = 2

6-Coloring Algorithm Simple minimum-degree Greedy algorithm

Find a vertex v of degree 5 or less. Remove v; the remaining graph G’ is still planar Inductively G’ with 6 colors Finally, color v with a color not used by its at most

5 neighbors in G’.

v

G

G’

Wireless transmissions

Channel A

Channel B

Hidden Collision if A=B

1

2

3

Primary Collision if A=B

Channel A

Channel B

1

2

Channel = time slot (TDMA), frequency (FDMA), etc.

Distance-2 Coloring Problem

Vertices with common neighbor must also receive different colors

D2-Col: Given: Graph G Find: Mapping : V(G) {1,2,…,} s.t.

Distance(u,v) 2 (u) (v)

2(G) = minimum number of colors needed in a distance-2 coloring of G

2(G) can get arbitrarily large

2(G) +1, for any G = max degree of G

So, no 4-color theorem, even for trees.

Any upper bound will be a function of

Wegner’s Conjecture [1973]

Conjecture: For a planar graph G, 2(G) 1.5(when 8)

1.5 colors can be necessary

/2

Results on d2-coloring planar graphs 9 [Ramanathan, Lloyd, ‘93] 8 - 22 [Jonas, ’93] 3+8 [Jendrol, Skupien, ’01] 2+25 [van den Heuvel, McGuinness, ’99] 1.8+1 [Agnarsson, H ’00] (for large)

Tight bound on a greedy algorithm do, [Borodin, et al, ’01] (for ≥ 47) 1.66 + 78 [Molloy, Salavatipour ’02]

Distance-2 Independent Set

D2-IS: set of vertices so that any pair is of distance more than 2

2(H) = largest size of a D2-IS in graph H

Distance-2 Independent Set

D2-IS: set of vertices so that any pair is of distance more than 2

2(H) = largest size of a D2-IS in graph H

Work in progress: Resolving Wegner’s

Conjecture Conjecture: For a planar graph G,

2(G) 1.5 Best upper bound known is

1.66 + O(1) So, Max D2-Indep. Set:

2(H) n / 1.66

2(H) = 1.5 Max D2-Indep. Set:

2(H) = n / 1.5

/2H

D2-Independent Set in planar graphs For a planar graph G,

2(G) n / 1.5 (1-o(1)) [H, ‘05] Can approximate 2(G) within 1+O(1/1/3)

Can color a (1-o(1)) fraction of the vertices using 1.5 colors.

Planar graph G

Planar graph G

Vh = High-degree vertices: degree 1/3 = b

Note: |Vh| n/b, which is negligible

Planar graph G

May assume that every low-degree vertex has at least 2 high-degree neighbors.

Otherwise, we can reduce those vertices.

d2-neighborhood of size at most + 2/3

Planar graph G

Eliminate lo-degree vertices with 3+ hi-degree neighbors

(total 3|Vh|)

Planar graph G’

Eliminate lo-degree vertices with “remote” neighbors

(again |Vh|)

Derived multigraph H

Form a multigraph H on nodes Vh, edge for each degree-2 vertex in G’

(show here the multiplicities in numbers)

6

6

6

5

6

Multigraph H

Find a maximum matching on H

6

6

6

5

6

Input graph G

Maximum matching on H: A d2-IS in G

Input graph G

Can also edge-color H: + 1.1-approximation of Nishizeki & Kashiwagi ’90

+ 1+eps-approximation, Sanders, SODA ‘05

H amounts to all but a O(1/b)-fraction of the graph

Summary

We have considered the Distance-2 coloring problem

Viewed some results on planar graphs

Questions ?

Complexity of Graph Coloring Decision problem

“Is 2(G) k” is NP-complete Even for planar graphs! Seek instead good approximations

Polynomial time algorithms Performance ratio of algorithm A:

)G(

)G(Amax

n|V|

Approximations of Graph Coloring [Feige, Kilian ’98] Graph Coloring is hard to

approximate within (n1-), for any >0 (assuming P NP)

[H ’93] Best performance ratio known is only O(n (loglog n)2 / (log n)3)

The correct exponent is 1 (modulo lower order terms)