Graph coloring: From maps to wireless networks and job scheduling Magnús 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 Guthrie & DeMorgan Guthrie & DeMorgan first proofs announced  Kempe + Tait 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 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 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   /2 H

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

V h = High-degree vertices: degree   1/3 = b Note: |V h |  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|V h |)

Planar graph G’ Eliminate lo-degree vertices with “remote” neighbors (again  |V h |)

Derived multigraph H Form a multigraph H on nodes V h, edge for each degree- 2 vertex in G’ (show here the multiplicities in numbers)

Multigraph H Find a maximum matching on H

Input graph G Maximum matching on H: A d2-IS in G

Input graph G Can also edge-color H: approximation of Nishizeki & Kashiwagi ’ 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:

Approximations of Graph Coloring [Feige, Kilian ’98] Graph Coloring is hard to approximate within  (n 1-  ), 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)