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CSC1300–DiscreteStructures 17:GraphColoring
DrPapalaskari 1
GraphColoring
CSC1300–DiscreteStructuresVillanovaUniversity
VillanovaCSC1300-DrPapalaskari 1
MajorThemes• Vertexcoloring• ChromaFcnumberχ(G)• Mapcoloring• Greedycoloringalgorithm• ApplicaFons
VillanovaCSC1300-DrPapalaskari 2
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
DrPapalaskari 2
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
VillanovaCSC1300-DrPapalaskari 11
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
DrPapalaskari 3
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
DrPapalaskari 4
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
DrPapalaskari 5
Earlierexample–seenasschedulingconstraints
VillanovaCSC1300-DrPapalaskari 21
CSC1700
CSC2014 CSC4480CSC2053 CSC2400
CSC1300 CSC1052
RevisedExamSchedule:
VillanovaCSC1300-DrPapalaskari 23
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
DrPapalaskari 6
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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