CSC1300–DiscreteStructures 17:GraphColoring

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GraphColoring

CSC1300–DiscreteStructuresVillanovaUniversity

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MajorThemes•  Vertexcoloring•  ChromaFcnumberχ(G)•  Mapcoloring•  Greedycoloringalgorithm•  ApplicaFons

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VertexColorings

4Source:“DiscreteMathemaFcs”byChartrand&Zhang,2011,WavelandPress.

ChromaFcnumberχ(G)=leastnumberofcolorsneededtocolortheverFcesofagraphsothatnotwoadjacentverFcesareassignedthesamecolor?

Adjacentver,cescannothavethesamecolor

5Source:“DiscreteMathemaFcswithDucks”bySara-MarieBelcastro,2012,CRCPress,Fig13.1.

WhatistheleastnumberofcolorsneededfortheverFcesofthisgraphsothatnotwoadjacentverFceshavethesamecolor?

χ(G)=

CSC1300–DiscreteStructures 17:GraphColoring

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MapColoringRegionèvertexCommonborderèedge

G

B

AC D

E

F

IG H

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MapColoring

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Whatistheleastnumberofcolorsneededtocoloramap?

ColoringtheUSA

hcp://people.math.gatech.edu/~thomas/FC/usa.gif

hcp://www.printco.com/pages/State%20Map%20Requirements/USA-colored-12-x-8.gif

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Fourcolortheorem

Every planar graph is 4-colorable

TheproofofthistheoremisoneofthemostfamousandcontroversialproofsinmathemaFcs,becauseitreliesonacomputerprogram.Itwasfirstpresentedin1976.AmorerecentreformulaFoncanbefoundinthisarFcle:FormalProof–TheFourColorTheorem,GeorgesGonthier,NoFcesoftheAmericanMathemaFcalSociety,December2008.hcp://www.ams.org/noFces/200811/tx081101382p.pdf

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CSC1300–DiscreteStructures 17:GraphColoring

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Doyoualwaysneedfourcolors?

Fourcolortheorem

Every planar graph is 4-colorable

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Whataboutnon-planargraphs?

Fourcolortheorem

Every planar graph is 4-colorable

K5K3,3

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Example

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Example

VillanovaCSC1300-DrPapalaskari 16Source:“DiscreteMathemaFcswithDucks”bySara-MarieBelcastro,2012,CRCPress,Fig13.1.

CSC1300–DiscreteStructures 17:GraphColoring

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Example

VillanovaCSC1300-DrPapalaskari 17Source:“DiscreteMathemaFcswithDucks”bySara-MarieBelcastro,2012,CRCPress,Fig13.1.

ChromaFcNumbersofSomeGraphs•  χ(G)=1iff...

•  ForKn,thecompletegraphwithnverFces,χ(Kn)=Corollary:IfagraphhasKnasitssubgraph,thenχ(Kn)=•  ForCn,thecyclewithnverFces,χ(Cn)=•  ForanybiparFtegraphG,χ(G)=•  ForanyplanargraphG,χ(G)≤4(FourColorTheorem)

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•  mapcoloring

•  scheduling– eg:Finalexamscheduling

•  FrequencyassignmentsforradiostaFons•  IndexregisterassignmentsincompileropFmizaFon

•  Phasesfortrafficlights

Applica,onsofGraphColoring

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Example:Scheduletheseexams,avoidingconflicts

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CSC1700CSC2014

CSC4480

CSC2053

CSC2400

CSC1300CSC1052

Monday Tuesday Wednesday

CSC1300–DiscreteStructures 17:GraphColoring

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Earlierexample–seenasschedulingconstraints

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CSC1700

CSC2014 CSC4480CSC2053 CSC2400

CSC1300 CSC1052

RevisedExamSchedule:

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CSC1700

CSC2014

CSC4480

CSC2053

CSC2400

CSC1300

CSC1052

Monday Tuesday Wednesday

??

Graphcoloringalgorithm?

VillanovaCSC1300-DrPapalaskari 24Source:“DiscreteMathemaFcswithDucks”bySara-MarieBelcastro,2012,CRCPress,p374.

Compu,ngtheChroma,cNumber

Thereisnoefficientalgorithmforfindingχ(G)forarbitrarygraphs.MostcomputerscienFstsbelievethatnosuchalgorithmexists.

Greedyalgorithm:sequen7alcoloring:1.  OrdertheverFcesinnonincreasingorderoftheirdegrees.2.  Scanthelisttocoloreachvertexinthefirstavailablecolor,i.e.,

thefirstcolornotusedforcoloringanyvertexadjacenttoit.

hcp://upload.wikimedia.org/wikipedia/commons/0/00/Greedy_colourings.svg

NotalwaysopFmal!(ordermacers)

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CSC1300–DiscreteStructures 17:GraphColoring

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Example:IndexRegisters

source:hcp://www.lighterra.com/papers/graphcoloring/

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AnotherApplicaFonofvertexcoloring:Trafficlights

•  seealsoexample13.3.9&Figure13.12

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AnotherApplicaFonofvertexcoloring:Trafficlights

•  seealsoexample13.3.9&Figure13.12

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