AQR UNIT 7

NETWORKS AND GRAPHS:

Coloring

Packet #4

BY:_____________________

Herearethreedogsandthreehouses.

Canyoufindapathfromeachdogtoeachhousesuchthatnotwopathsintersect?_________________________________________________________________________________________________________________ ColoringaMAP:

MapA MapB MapC

5 Graph Theory II

Dog DogDog

Can you find a path from each dog to each house such that no two paths intersect?

A quadapus is a little-known animal similar to an octopus, but with four arms. Here are five quadapi resting on the seafloor:

Can each quadapus simultaneously shake hands with every other in such a way that no arms cross?

Informally, a planar graph is a graph that can be drawn in the plane so that no edges cross. Thus, these two puzzles are asking whether the graphs below are planar; that is, whether they can be redrawn so that no edges cross.

In each case, the answer is, “No— but almost!” In fact, each drawing would be possible if any single edge were removed.

More precisely, graph is planar if it has a planar embedding (or drawing). This is a way of associating each vertex with a distinct point in the plane and each edge with a continuous, non-self-intersecting curve such that:

• The endpoints of the curve associated with an edge (u,�v)�are the points associated with vertices u�and v.

Teacher Version

Networks and Graphs: Graph Coloring

VII.C Student Activity Sheet 9: Map Coloring

Charles A. Dana Center at The University of Texas at Austin

Advanced Mathematical Decision Making (2010)

Activity Sheet 9, 3 pages

VII-76

Map Coloring Problem

You are the publisher of a new edition of the world atlas. As you prepare the different maps for printing, you need to make sure that countries adjacent to each other (sharing a common border) are given different colors.

1. For the following two maps, decide how to color each of the five countries (regions) so that no two adjacent countries are colored the same. Treat the outside region as a single country (perhaps it represents an ocean colored blue). Assume that every country is composed of a single contiguous region (for example, you treat Alaska and Hawaii as separate regions when constructing a map of the world).

Map I, Solution A: Color each region a different color.

Map I, Solution B: Regions 2 and 4 can be the same color, while the other regions are all different colors.

Map II, Solution A: Color each region a different color.

Map II, Solution B: Regions 1 and 3 can be the same color, while the other regions are all different colors (Or Regions 1 and 4 can be the same. Or Regions 2 and 4 can be the same). Regions 1 and 3 only meet at a single point, so they can be the same color.

Map II, Solution C: Use one color for Regions 1 and 3, another color for Regions 2 and 4, and a third color for Region 5.

Map I

Map II

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Teacher Version

Networks and Graphs: Graph Coloring

VII.C Student Activity Sheet 9: Map Coloring

Charles A. Dana Center at The University of Texas at Austin

Advanced Mathematical Decision Making (2010)

Activity Sheet 9, 3 pages

VII-76

Map Coloring Problem

You are the publisher of a new edition of the world atlas. As you prepare the different maps for printing, you need to make sure that countries adjacent to each other (sharing a common border) are given different colors.

1. For the following two maps, decide how to color each of the five countries (regions) so that no two adjacent countries are colored the same. Treat the outside region as a single country (perhaps it represents an ocean colored blue). Assume that every country is composed of a single contiguous region (for example, you treat Alaska and Hawaii as separate regions when constructing a map of the world).

Map I, Solution A: Color each region a different color.

Map I, Solution B: Regions 2 and 4 can be the same color, while the other regions are all different colors.

Map II, Solution A: Color each region a different color.

Map II, Solution B: Regions 1 and 3 can be the same color, while the other regions are all different colors (Or Regions 1 and 4 can be the same. Or Regions 2 and 4 can be the same). Regions 1 and 3 only meet at a single point, so they can be the same color.

Map II, Solution C: Use one color for Regions 1 and 3, another color for Regions 2 and 4, and a third color for Region 5.

Map I

Map II

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Createamapthatneedsfivecolors.Whatisthelargestnumberofcolorsrequiredtocoloranymap,thatkeepsadjacentregionsseparate?Justifyyourresponse.

_________________________________________________________________________________________________________________ MapsbecomingGraphs:

Revisitthemapcoloringexercisesfromaboveintermsofgraphs.Forexample,MapIcanberepresentedbythefollowinggraph.Thegraphshouldincludeavertexforeachcountry(orregion)inyourmap.Iftwocountriesshareaborderandneedtobecoloreddifferently,thegraphshowsanedgebetweentheverticesthatrepresentthem.AfterstudyingtherelationshipbetweenMapIandthegraphforMapI,createagraphthatrepresentsMapII.

Teacher Version

Networks and Graphs: Graph Coloring VII.C Student Activity Sheet 9: Map Coloring

Charles A. Dana Center at The University of Texas at Austin

Advanced Mathematical Decision Making (2010)

Activity Sheet 9, 3 pages

VII-77

2. How many colors did you use to color each map?

Map I, Solution A uses five colors.

Map I, Solution B uses four colors.

Map II, Solution A uses five colors.

Map II, Solution B uses four colors.

Map II, Solution C uses three colors.

3. REFLECTION: Did you use fewer colors than anyone else? If not, describe how you can adjust your map to use fewer colors. If yes, how are you confident that the fewest colors have been used that can be?

Responses to this Reflection will show students’ different approaches to the question.

4. If you want to color each map using the least number of colors (still keeping adjacent regions separate colors), how many colors are needed for each map?

Map I needs four colors.

Map II needs three colors. 5. Create a map that requires the use of three colors.

Answers will vary. Map II is an example of a map that requires three colors.

Teacher Version

Networks and Graphs: Graph Coloring VII.C Student Activity Sheet 9: Map Coloring

Charles A. Dana Center at The University of Texas at Austin

Advanced Mathematical Decision Making (2010)

Activity Sheet 9, 3 pages

VII-78

6. Create a map with at least four different regions that could be colored with two colors.

Starting with the inner country, you can alternate between two colors as you color the outer regions. Even though some countries are completely enclosed by others, every country (or region) is connected to at least one other. This satisfies the requirements for a map.

7. EXTENSION: Create a map that needs five colors. What is the largest number of colors required to color any map, that keeps adjacent regions separate? Justify your response.

Answers will vary. As long as every country is connected, you never need to use five colors. Four colors always suffice.

Teacher Version

Networks and Graphs: Graph Coloring

VII.C Student Activity Sheet 10: Coloring Maps and Scheduling

Charles A. Dana Center at The University of Texas at Austin

Advanced Mathematical Decision Making (2010)

Activity Sheet 10, 8 pages

VII-79

Creating Graphs from Maps

1. Revisit the map coloring exercises from Student Activity Sheet 9 in terms of graphs. For

example, Map I can be represented by the following graph. The graph should include a

vertex for each country (or region) in your map. If two countries share a border and need

to be colored differently, the graph shows an edge between the vertices that represent

them.

After studying the relationship between Map I and the graph for Map I, create a graph

that represents Map II.

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Whenmightagraphnotcorrespondtoamap?Whatdoesthemaplooklikeforthisgraph?

__________________________________________________________________________________________________________________ColoringaGraph:Coloreachvertexsothatconnectedverticeshavedifferentcolors

Teacher Version

Networks and Graphs: Graph Coloring

VII.C Student Activity Sheet 10: Coloring Maps and Scheduling

Charles A. Dana Center at The University of Texas at Austin

Advanced Mathematical Decision Making (2010)

Activity Sheet 10, 8 pages

VII-79

Creating Graphs from Maps

1. Revisit the map coloring exercises from Student Activity Sheet 9 in terms of graphs. For

example, Map I can be represented by the following graph. The graph should include a

vertex for each country (or region) in your map. If two countries share a border and need

to be colored differently, the graph shows an edge between the vertices that represent

them.

After studying the relationship between Map I and the graph for Map I, create a graph

that represents Map II.

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5 Teacher Version

Networks and Graphs: Graph Coloring VII.C Student Activity Sheet 10: Coloring Maps and Scheduling

Charles A. Dana Center at The University of Texas at Austin

Advanced Mathematical Decision Making (2010)

Activity Sheet 10, 8 pages

VII-82

7. REFLECTION: When might a graph not correspond to a map?

Answers will vary. Sample student response:

If a graph can be drawn without any intersecting edges, it represents a map. The graph in Question 6 has numerous edges that intersect. Some of these intersections can be avoided, but not all. The best you can do is as follows:

The last edge (represented with a dashed line) cannot be added without creating a crossing. Interestingly enough, this is exactly the same reason that makes it impossible to create a corresponding map for this graph.

8. The chromatic number of a graph is the minimum number of colors needed to color each vertex in such a way that any two vertices sharing an edge are a different color. Provide

examples of graphs that have chromatic numbers of 3 and 4.

Graph I has a chromatic number of 3, since each vertex is adjacent to the other two. Graph II has a chromatic number of 4, since each vertex is adjacent to the other three.

Graph I Graph II

Thechromaticnumberofagraphistheminimumnumberofcolorsneededtocoloreachvertexinsuchawaythatanytwoverticessharinganedgeareadifferentcolor.

Createanothergraphthatcouldbecoloredwithtwocolors.Determinewhattypesofgraphscanalwaysbecoloredwithtwocolors.

Student Worksheets Created by Matthew M. Winking at Phoenix High School SECTION 7-5 p.94!

!

Convert the following map in to a graph (regions become vertices).

Determine the Chromatic number of each of the following graphs.

A"I"G"

H"E"

B"C"D"

F"

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G

__________________________________________________________________________________________________________________GreedyColoringAlgorithm:Thealgorithmiscalledgreedybecauseitisarathershort-sightedwayoftryingtomakeapropercoloringwithasfewcolorsaspossible.Itdoesnotalwayssucceedinfindingtheminimum2number(thechromaticnumber),butatleastprovidessomepropercoloring.Theprocedurerequiresustonumberconsecutivelythecolorsthatweuse,soeachtimeweintroduceanewcolor,wenumberitalso.Hereistheprocedure:1.Coloravertexwithcolor1.2.Pickanuncoloredvertex.Coloritwiththelowest-numberedcolorthathasnotbeenusedonanypreviously-coloredverticesadjacenttoit.(Ifallpreviously-usedcolorsappearonverticesadjacenttov,thismeansthatwemustintroduceanewcolorandnumberit.)3.Repeatthepreviousstepuntilallverticesarecolored.Clearly,thisproducesapropercoloring,sincewearecarefultoavoidconflictseachtimewecoloranewvertex.Howmanycolorswillbeused?Itishardtosayinadvance,anditdependsonwhatorderwechoosetocolorthevertices.Nonetheless,thereisacertainminimumqualityweget,whichwecandeterminebythefollowingtheoreticalargument:Supposethatdisthelargestdegreeofanyvertexinourgraph,i.e.,allverticeshavedorfeweredgesattached,andatleastonevertexhaspreciselydedgesattached.Aswegoaboutcoloring,whenwecoloranyparticularvertexv,itisattachedtoatmostdothervertices,ofwhichsomemayalreadybecolored.Thenthereareatmostdcolorsthatwemustavoidusing.Weusethelowest-numberedcolornotprohibited.Thatmeansthatweusesomecolornumberedd+1orlower,becauseatleastoneofthecolors1,2,...,d+1isNOTprohibited.Soweneverneedtouseanycolornumberedhigherthand+1.Thisgivesusthefollowingtheorem:GreedyColoringTheorem! IfdisthelargestofthedegreesoftheverticesinagraphG,thenGhasapropercoloringwithd+1orfewercolors,i.e.,thechromaticnumberofGisatmostd+1.Tryitonthisgraph:

__________________________________________________________________________________________________________________Scheduling:Mrs.Jacobs,thenewprincipalatRiverdaleHighSchool,wantstomakeagoodimpressionbyofferingalotofnewexcitingclassesforherstudents.Theprincipalplanstouseherknowledgeofgraphtheorytodeterminewheneachclasswillbeoffered.Sincesheistryingtomakeherstudentshappy,Mrs.Jacobsdoesnotwanttooffertwodifferentclassesatthesametimeiftherearestudentswantingtotakeboth.Shedecidestoconstructagraphinthefollowingway:Eachclassisrepresentedbyavertexandifthereisastudentinterestedintwoclasses,thosetwoverticesareconnectedbyanedge.Supposetherearefiveclasses(A,B,C,D,andE)andonlyfivestudentswishingtotakethefollowingclasses:

1. JasonwantstotakeClassesAandE.2. EmorywanttotakeClassesB,C,andE.3. FelicitywantstotakeClassesAandD.4. GeoffwantstotakeClassesBandC.5. HilarywantstotakeClassesDandE.

Constructthegraphfortheprincipal.Findthechromaticnumberofthegraph,andcolorthegraphusingtheleastnumberofcolors.__________________________________________________________________________________________________________________Follow-upQuestions:1)Ifyouwanttocoloreachmapusingtheleastnumberofcolors(stillkeepingadjacentregionsseparatecolors),howmanycolorsareneededforeachmap?2)RestatethefollowingMapColoringproblemintermsofaGraphColoringproblem.Youarethepublisherofaneweditionoftheworldatlas.Asyoupreparethedifferentmapsforprinting,youneedtomakesurethatcountriesadjacenttoeachother(sharingacommonborder)aregivendifferentcolors.

Restate!3)Howcanthegraphcoloringsolutionhelptheprincipalwithherschedulingproblem?4)Thinkaboutagraphwith20verticesthathasachromaticnumberof2.Doesyourgraphhaveanycycles?(Recall:Acycleisapaththroughthegraphthatstartsandendsatthesamevertexanddoesnotreuseanyedges.)________________________________________________________________________________________________________________Problems:1)Createanew(differentfromyourothers)mapthatrequirestheuseofthreecolors.2)Createagraphthatrequiresthreecolors.3)Createagraphthatneedsfivecolors,andthendrawtheassociatedmap.4)Provideexamplesofgraphsthathavechromaticnumbersof3and4.5)Colorthisgraphwiththeminimumnumberofcolors.