Chromatic numbers for the hyperbolic plane and discrete analogs

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by  Hugo Parlier

Chromatic numbers for the hyperbolic plane and discrete analogs
HugoParlier**,CamillePetit****
Abstract. Westudycoloringsofthehyperbolicplane,analogouslytotheHadwiger-Nelson
problemfortheEuclideanplane. Theideaistocolorpointsusingtheminimumnumber
ofcolorssuchthatnotwopointsatdistanceexactlydareofthesamecolor. Theproblem
dependsondand,followingastrategyofKloeckner,weshowlinearupperboundsonthe
7 necessarynumberofcolors. Inparallel,westudythesameproblemonq-regulartreesand
1
0 show analogous results. For both settings, we also consider a variant which consists in
2
replacingdwithanintervalofdistances.

n
a
J

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1. Introduction

]
O
H
Thegeometryofthehyperbolicplane appearsinalargevarietyofmathematicalcontexts
C
and,assuch,hasbeenextensivelystudied. Nonetheless,therearecertaincombinatorial
.
h
H
t questionsabout aboutwhichnotmuchisknown. We’remainlyinterestedinatypeof
a
m chromaticnumberforH.
[
ThecelebratedHadwiger-Nelsonproblemisthesearchfortheminimalnumberofcolors
1
v necessarytocolortheEuclideanplanesuchthatanytwopointsatdistance1arecolored
8 differently. Thischromaticnumber,denotedbyχ(R2),hasbeenknowntobetween4and
4
6 7forahalf-century,butsignificantprogresshaseludedmathematiciansfordecades(see
8
0 [55] for details). This can - and has - been studied for other metric spaces such as Rn [44].
.
1 Thechoiceofdistance1foraEuclideanspaceisnotimportantthankstohomotheties. In
0
7 generalhowever,thechromaticnumberofametricspacewilldependonachoiceofd > 0
1
andcoloringsarerequiredtohavepointsatdistanceexactlydcoloreddifferently.
:
v
Xi For H the choice of d is important and we denote the d-chromatic number χ(H,d). As
r suggestedin[22],lettingdgrowandstudyingthegrowthofχ(H,d)couldbecomparedto
a
thestudyofχ(Rn)forgrowingnwhichisknowntogrowexponentiallyinn(see[11,44]and
referencestherein). Theanalogywillonlybeinterestingifχ(H,d)isshowntogrowwith
d but that’s not known to be true. The same proof as for the Euclidean plane [22] gives a
*ResearchsupportedbySwissNationalScienceFoundationgrantnumberPP00P2 153024
**ResearchsupportedbySwissNationalScienceFoundationgrantnumber200021 153599
2010MathematicsSubjectClassification: Primary: 05C15,30F45. Secondary: 05C63,53C22,
30F10.
Keywordsandphrases: chromaticnumbers,hyperbolicplane,trees
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universallowerboundof4forχ(H,d)andthatseemstobetheextentofthecurrentstate
ofknowledgeforlowerbounds.
Ourfocuspointwillbeonupperbounds. Thefollowingtheoremsummarizessomeofour
concreteresults.
Theorem1.1. Ford ≤ 2log(2) ≈ 1.389...wehave
χ(H,d) ≤ 9.
Ford ≤ 2log(3)
χ(H,d) ≤ 12.
Ford ≥ 2log(3)thefollowingholds:
(cid:18)(cid:24) (cid:25) (cid:19)
d
χ(H,d) ≤ 5 +1 .
log(4)
Ourmethodsandprooffollowthegeneralstrategyofusinga”hyperboliccheckerboard”,a
methodoutlinedin[22]andattributedtoSze´kely. Kloeckner[22]explainshowtogetalinear
upperbound(ind)andasksmanyinterestingquestions. Ourboundsansweroneofthe
questions(ProblemR).Moreimportantly,weoptimizethestrategy(Theorems33..22and33..33)
andprovidesomemissingarguments. Itistheseadditionaldetailsthatallowforimproved
bounds for both small d and larger d (Theorems 33..44, 33..66 and Proposition 33..55). Note that
thesequestionscouldalsobeaskedmoregenerallyforanyhyperbolicsurface,but,aswas
shownbytheauthorsin[33],theboundsareverydifferentandgrowexponentiallyind.
We note that for small d, it seems very unlikely that the bounds we provide are close to
optimal. ThisisillustratedinProposition33..88whereweshowhowtouseafundamental
domaintoboundχ(H,d)by8,butitonlyworksforcertainvaluesofd.
Whenstudyingtheproblemofthehyperbolicplane,westartedlookingfordiscreteanalogs
thatmighthelpusunderstandthestructureofsubgraphsofHthatoccurforlargerdand
thathavehyperbolicityproperties. Thisleadustolookingatinfinite q-regulartrees. Al-
thoughtheyarebipartite,wecanlookattheird-chromaticnumberandstudyitanalogously
H H
to . Theupperboundsweobtainedarecloseinspirittothoseof andobtainedbya
similarmethod. Wesynthesizethemasfollows(seeTheorem33..1100).
Theorem1.2. Ifdisoddthen
χ(T ,d) = 2.
q
Ifdiseventhen
χ(T ,d) ≤ (q−1)(d+1).
q
2
H
Lowerboundsseemdifficult,justlikefor . Onewayofobtaininglowerboundsisusinga
typeofcliquenumber,herethemaximalnumberofpointsatpairwisedistanced. Foreven
dthiscliquenumberisalwaysq(seeProposition33..1111)whichisquitefarfromourupper
bounds. We can improve on that, but only slightly, by producing a generalized Moser
spindle(Proposition33..1133). Thisgivesalowerboundofq+1. Nonetheless,weknowthis
lowerboundisnotoptimalasbyanextensivecomputersearchwefoundthatχ(T ,8) ≥ 5
3
H
(seeRemark33..1144). Asfor ,thecombinatoricsseemtogetoutofhandprettyquickly.
A property shared by both H and T is that both are natural homogeneous Gromov
q
hyperbolicspaces. Inparticular,theyhavethintrianglesbywhichwemeanthatgeodesic
triangles with long sides look roughly like tripods (and for T they are tripods). This
q
suggeststhatanintervalchromaticproblemmightberelevant. Inthisadaptation,wefix
aninterval[d,cd]withd > 0andc > 1. Weaskthatpointsthathavedistancesthatliein
[d,cd]becoloreddifferently.
Kloeckner[22]pointsoutthatfortheEuclideanplanethisintervalchromaticnumbergrows
likec2 forfixeddandgrowingcandaskswhether
χ(R2,[d,cd])
lim
c→∞ c2
exists. Hestatesapurposefullyvagueintervalchromaticproblemforthehyperbolicplane
(ProblemZfrom[22]). We’reabletoshowthefollowingresults(Theorems44..11and44..22):
Theorem1.3. Forsufficientlylarged,thequantityχ(H,[d,cd])satisfies
(cid:16) (cid:17)
2ecd2−1 < χ(H,[d,cd]) < 2 2ecd2−1 +1 (cd+1).
For T ,usingthesametechniques,weshowthefollowing(Theorems44..33and44..44).
q
Theorem1.4. Thequantityχ(T ,[d,cd])satisfies
q
q(q−1)(cid:98)c2d(cid:99)−(cid:100)d2(cid:101) ≤ χ(Tq,[d,cd]) ≤ (q−1)(cid:98)c2d+1(cid:99)((cid:98)cd(cid:99)+1).
Thelowerboundsinboththeoremsabovecomefromlowerboundsonthe(interval)clique
numbers.
Acknowledgements.
WeheartilythankAmmarHalabiforgraciouslywritingthecodenecessarytotestchromatic
numbersforregulartrees. Remark33..1144isthankstohim.
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2. Preliminaries
2.1. Chromaticnumbersofmetricspaces
ThechromaticnumberofagraphGistheminimalnumberofcolorsneededtocolorthe
verticesofagraphsuchthatanytwoadjacentverticesareofdifferentcolors.
Given a metric space (X,δ) and a number d > 0, we define the chromatic number
χ((X,δ),d) relative to d > 0 to be the minimal number of colors needed to color all
pointsof X suchthatany x,y ∈ X withδ(x,y) = darecoloreddifferently. We’llsometimes
refertothed-chromaticnumberof(X,δ). Onecandefinethechromaticnumberofametric
spaceviathechromaticnumberofgraphsasfollows. Givenametricspace (X,δ) anda
realnumberd > 0,weconstructagraphG({X,δ},d)withverticespointsofXandanedge
betweenpointsiftheyareexactlyatdistanced.
We’llrefertotheabovechromaticnumbersasbeingpurechromaticnumbers,asopposedto
thenotionwe’llintroducenow.
One variant on the pure chromatic number is to ask that points that lie at a distance
belongingtoagivensetbeofdifferentcolors. Anexampleofthisisthechromaticnumber
of Gk power of a graph G. This is equivalent to asking that any two vertices at distance
belonging to the set {1,...,k} be colored differently. More generally, for a metric space
(X,δ)andasetofdistances∆,the∆-chromaticnumberχ((X,δ),∆)istheminimalnumber
ofcolorsnecessarytocolorpointsof X suchthatanytwopointsatdistancebelongingto
∆
areofadifferentcolor. Asabove,thiscanbeseenasthechromaticnumberofagraph
G({X,δ},D)whereverticesarepointsof X andedgesbelongto D. We’llbeparticularly
interested in this problem when D is an interval [a,b]. We’ll refer to these quantities as
intervalchromaticnumbers.
A straightforward way of obtaining a lower bound for chromatic numbers of graphs is
via the clique number which is the order of the largest embedded complete graph. The
cliquenumberΩ(G)clearlysatisfiesΩ(G) ≤ χ(G). SimilarlywedefineΩ((X,δ),d),resp.
Ω((X,δ),∆), to be the size of the largest number of points of X all pairwise at distance
exactlyd,resp. allatdistancelyingin∆.
2.2. Ourmetricspaces
Thetwotypesofmetricspaceswe’llworkwitharethehyperbolicplaneHandq-regular
trees (for q ≥ 3). The unique infinite tree of degree q in every vertex will be denoted T .
q
H
Bothareviewedasmetricspace, withthestandardPoincare´ metric(anexplicitdistance
formulawillbeprovidedbelow)andT asametricspaceonverticesobtainedbyassigning
q
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length1toeachedge. Althoughwethinkofregulartreesasatypeofdiscreteanalogof
thehyperbolicplane,notethatthetwometricspacesarenotevenquasi-isometrictoone
another.
Inthenextsectionwe’llbrieflydescribeametricrelationshipbetweenHand T ,namelya
q
quasi-isometricembeddingof T intoH. Itisprovidedformotivationalpurposesand,asit
q
willnotbeusedinthesequel,itcanbeskippedbythelessinterestedreader.
2.3. Locallyflatmodelsofthehyperbolicplaneandgeometricallyembeddedtrees
H
Wedescribealocally”flat”modelofthehyperbolicplanewhichisquasi-isometricto
intowhichregulartreesgeometricallyembed.
Onewayofconstructingaspacewhichsharespropertieswithhyperbolicplaneistopaste
togethercopiesofanequilateralEuclideantriangleτ withsideslengths1.
Todoso,fixanintegern ≥ 6andconstructasimplyconnectedspaceasfollows. Starting
with a base copy of τ, paste n copies of τ around each vertex to obtain a larger simply
connected shape. Then we repeat the process indefinitely to get an unbounded simply
connecteddomainwhichwe’lldenote H .
n
Forexample: ifn = 6thentheresultistheEuclideanplane. Inparticular,verticesofcopies
ofτ maptopointsofangle2π.
However,foranyn ≥ 7,thesetofverticesmapstosingularpointsofangle πn. Wenote
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that,foralln ≥ 6, H isaCAT(0)metricspace.
n
Thefollowingiswell-knowntoexperts,butweprovideasketchproofforcompleteness.
Proposition2.1. Forn ≥ 7, H andHarequasi-isometric.
n
Proof. Toseethis,itsufficestoconstructaquasi-isometrybetweenHand H . Considerthe
n
celldecompositionof H dualtoitstriangulation: eachcellisann-gonwithasingularity
n
in its center. As it is dual to a triangulation, the valency in each vertex of this n-gon
decompositionisthree. Denoteby P acopyofthissingularn-gon.
n
Now consider the unique tiling (up to isometry) of H by regular n-gons of angles 2π.
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DenotebyQ thishyperbolicn-gonweusetotile. Notethatboth P andQ havethen-th
n n n
dihedralgroup D asisometrygroup.
n
Let f : H → Q be any bijective map which, for simplicity, we’ll suppose sends the
n n
boundarytotheboundaryandisinvariantbytheactionsofD . (Thisisactuallynotstrictly
n
necessarybutitsimplifiesthediscussionsomewhat.) Themap f,bycompactnessof H
n
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andQ ,isofboundeddistortion.
n
TherearenownaturalmapsbetweenHand H whichconsistsonreplacingeachregular
n
hyperbolicn-gonofthetilingbythesingularEuclideananalogueandvice-versa. Pointsare
associatedvia f andbyinvarianceof D ,coincidewithrespecttothepasting. Theresultis
n
abijectionbetweenHand H whichisclearlyofboundeddistortion.
n
Thereasonwe’veintroduced H isthatwehavethefollowingembedding. Byisometric
n
embeddingwemeananembeddingbetweenmetricspaces(X ,δ ) (cid:44)→ (X ,δ )suchthat
1 1 2 2
theinducedmetricon X by X coincideswiththemetricδ .
1 2 1
Proposition2.2. Foranyq ≤ (cid:98)n(cid:99), T isometricallyembedsinto H .
3 q n
Proof. Therearetwothingstoprove. Thefirstisthatthereisanembedding. Todoso,we
thinkof T asbeingembeddedintheplane(thisgivesusanorientationateveryvertex).
q
Byverticesof H wemeanthesetofpointsthataretheimageoftheverticesofthetriangles
n
usedtoconstruct H . Byedgesof H ,wemeantheimageoftheedgesofthetriangles(all
n n
oflength1). We’regoingtomapverticesandedgesof T totheircounterpartsin H
q n
Takeabasevertexv of T andmapittoabasevertexw of H . Nowmapanedgeeof T
0 q 0 n q
incidenttov toanedgee(cid:48) of H incidenttow .
0 n 0
Wenowmapedgesincidentin v toedgesincidentin w . Edgesaround v and w both
0 0 0 0
haveorientationsandareorderedrelativelytoeande(cid:48). Followingthisorientation,edges
incidenttov aremappedtoedgesincidenttow ensuringthatifanytwoedgesin P that
0 0 n
areimageedgesformanangleofatleastπ. Inotherterms,followingtheorderaroundw ,
0
thereareatleasttwoedgesbetweenimageedges. Notethatthiswaspossiblethankstothe
conditiononqandn.
We’venowmappedalledgesofT incidentinv . Wenowrepeatthistomaptoallvertices
q 0
distance1fromv ,andtheninductively,tothoseatdistancer ≥ 2. Thisprovidesuswith
0
anembedding ϕ.
Byconstruction,theembeddingisgeometric. Indeed,letv,wbeverticesof ϕ(T )andletγ
q
betheuniquesimplepathbetweenthemcontainedin ϕ(T )(imageoftheuniquegeodesic
q
in T ). Wewanttocheckthatγislocallygeodesiceverywhere. Asthespace H isCAT(0),
q n
thiswillguaranteethatγistheuniquegeodesicin H betweenvandw. Todosowecheck
n
theangleconditionsalongγ. Byconstruction,theangleisπalongtheflatportionsofγand
atleastπ ineveryvertexbyconstruction. Thisprovesthatourembeddingisisometric.
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In terms of chromatic numbers, the isometric embedding above provides the following
immediatelowerbound.
Corollary2.3. Foralld ≥ 1andq ≥ 3andnsatisfyingn ≥ (cid:98)n(cid:99):
3
χ(T ,d) ≤ χ(H ,d).
q n
3. Purechromaticnumberproblem
Inthissectionwe’llbeconcernedwithfindingupperandlowerboundsforthed-chromatic
numberforboththehyperbolicplaneandforq-trees. Webeginwiththeformer.
3.1. Boundsforthehyperbolicplane
H
We’ll be using the upper half plane model of the hyperbolic plane. The hyperbolic
distanceformulaforH = {(x,y) ∈ R2 | y > 0}canbeexpressedas
dH(cid:0)(x,y),(x(cid:48),y(cid:48))(cid:1) = arccosh(cid:18)1+ (x−x(cid:48))2+(y−y(cid:48))2(cid:19).
2yy(cid:48)
WewanttominimallycolorHforgivend > 0suchthatanytwopointsatdistancedareof
differentcolors.
Onequickwordaboutlowerboundsforthesequantities. Gettingagoodlowerboundvia
inducedcompletegraphsisfutile-justlikefortheEuclideanplanethecliquenumberhas
anupperboundof3. AndjustlikeintheEuclideanplane,alowerboundof4foranydcan
beobtainedbyfindingametriccopyoftheMoserspindle. Itseemslikelythatonecando
better,atleastforlarged,butthecombinatoricsquicklygetoutofhand.
H
Sowefocusonupperbounds. Thegeneralstrategywillalwaysbethesame: cover with
monochromaticregionsofdiameterlessthandandensurethatanytworegionsofthesame
coloraresufficientlyfarapart.
3.1.1. Constructionofahyperboliccheckerboard
To color the hyperbolic plane we use a horocyclic checker board constructed as follows.
Accordingto[22]whereitisusedtocolorthehyperbolicplane,thisconstructionisdueto
Sze´kely. Unfortunatelysomeofthekeydetailsin[22]areincorrectsoforcompletenesswe
provideadetailedconstruction.
Themethodconsistsintilingthehyperbolicplanebyisometricrectangleswheretwosides
oftherectanglearesubarcsofgeodesicswithacommonpointatinfinity(socalledultra
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parallel curves) and the other two sides are horocyles surrounding that same point at
infinity.
Todosoformally,webeginbycuttingthehyperbolicplaneintoinfinitestripsboundedby
horocyclesasfollows. Wefixh > 0andset j ∈ Z,thestripS (h)tobetheset
j
(cid:110) (cid:111)
S (h) := (x,y) ∈ H | y ∈ [ejh,e(j+1)h[ .
j
We’llsometimescallS (h)astrata. Inthismodelthehorizontallinesy = ejh arehorocycles
j
around∞andhisthedistancebetweenthehorocyclesy = ejh andy = e(j+1)h.
Roughlyspeaking,wenowsubdividethestripsS (h)bycuttingthemalongverticallines
j
(geodesics). Wefixavaluew > 0andcutalongverticalgeodesicsegmentsinawaythat
the two endpoints of the base of each rectangle are at distance w. As we don’t want the
rectanglestooverlapevenintheirboundary,wechoosethattheverticalgeodesicsegments
belongstotherectangleonitsright. Thereissomechoiceisdoingtheaboveprocedurebut
wewillnotmakeuseofthatchoiceinanyway(andinfacttryingtousethishorizontal
parameteristricky). Onepossiblechoiceleadstothefollowingrectangleswhichforfixed
h,wwecanlabelwithelementsofZ2:
(cid:110) (cid:111)
R (h,w) := (x,y) ∈ H | x ∈ [rjeih,r(j+1)eih[, y ∈ [ejh,e(j+1)h[
i,j
where
(cid:113)
r := 2(cosh(w)−1).
(cid:16) (cid:17)
arccosh 1+ cosh(w)−1
e2h
(0,eh) (r,eh)
h
w
(0,1) (r,1)
Figure1: Therectangle R (h,w)
0,0
Iftheaboveformulaisslightlyconfusing,it’susefultokeepinmindoneparticularcopyof
therectanglesinceallofthemareisometric. Therectangle R (h,w)hasitsfourvertices
0,0
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givenbythepoints(0,1),(r,1),(0,eh)and(r,eh). Althoughthebasepointsoftherectangle
areatdistancew,theuppercornersareclosertoeachotherandtheirdistanceisinfact
(cid:18) (cid:19)
cosh(w)−1
arccosh 1+ .
e2h
Understanding the geometry of the rectangles is key in our argument and we want to
understandthediameterofthe(closed)rectangle.
Via a simple variational argument, the diameter is realized by some pair of points that
lieinthecorners. Asdiscussedabove,thedistancebetweentheuppercornersissmaller
thanthedistancebetweenpointsonthebasebutonepossibilityisthatthebottomcorners
realizethediameterandinfactforfixedwandsmallenoughh,thisisthecase. Theother
possibilityisthatoppositecornersrealizethediameterandtheirdistanceis
(cid:18) 2(cosh(w)−1)+(eh−1)2(cid:19)
arccosh 1+ .
2eh
Again,ifhissufficientlylarge,theabovevaluewillbethediameter.
To see the above observations, it suffices to look at the distance formula for a pair of
points(0,y)and(r,y(cid:48)). Wethinkofy(cid:48) asbeingavariablebeginningaty(cid:48) = y. Nowasy(cid:48)
increases,theirdistancebeginsbydecreasinguntileventuallyreachingaminimumand
then increasing towards infinity. Thus there is a certain value of y(cid:48) > y for which the
distancebetween(0,y)and(r,y(cid:48))isexactlythatofthedistancebetween(0,y)and(r,y).
Thisshowsthatthepairofpointsthatrealizethediameteriseitherthebaseorthediagonal
andthisdependsonhowlargehis. Allinall,we’veshownthefollowing.
Proposition3.1. Therectangle R (w,h)satisfies
i,j
(cid:0) (cid:1) (cid:26) (cid:18) 2(cosh(w)−1)+(eh−1)2(cid:19)(cid:27)
diam R (w,h) = max w, arccosh 1+ .
i,j 2eh
Wealsoneedtogetahandleonthedistancebetweenconsecutiverectanglesinastratum
(so rectangles Ri,j(w,h) and Ri(cid:48),j(w,h) for i(cid:48) > i). By the same considerations as above
thedistancewillberealized(intheclosureoftherectangles)bytheupperrightcornerof
Ri,j(w,h)andtheupperleftcornerof Ri(cid:48),j(w,h). Thedistanceformulagivesus
(cid:0) (cid:1) (cid:18) (i−i(cid:48))2(cosh(w)−1)(cid:19)
dH Ri,j(w,h),Ri(cid:48),j(w,h) = arccosh 1+ e2h .
With this in hand we can construct a well-adapted checker board in function of the pa-
rameter d. The method is to use a checkerboard with rectangles of diameter ≤ d. We
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color stratum by stratum cyclically using the above formula to ensure that if rectangles
arehorizontallysufficientlyfarapart,theycanbecoloredthesameway. Wethenneedto
repeat the above process with completely new colors until the strata are sufficiently far
apart(seeFigure22). Thisrequiresexactly(cid:100)d(cid:101)+1stratatobecoloredbeforerepeatingthe
h
procedure.
Thisleadstothefollowinggeneralstatementthatwestateasatheorem.
Theorem3.2. Thed-chromaticnumberofthehyperbolicplanesatisfies
(cid:18)(cid:24) (cid:25) (cid:19)
d
χ(H,d) ≤ (k+1) +1
h
foranyw,hthatsatisfy
(cid:26) (cid:18) 2(cosh(w)−1)+(eh−1)2(cid:19)(cid:27)
max w, arccosh 1+ ≤ d
2eh
andkisthesmallestintegersatisfying
(cid:115)
cosh(d)−1
k ≥ eh .
cosh(w)−1
Wenotethattheaboveconditionimpliesthatk ≥ 2.
≥ d
≥ d
(cid:108) (cid:109)
Figure2: k = 3and d = 2
h
Asstated,itisnotclearhowtooptimallyapplythetheorem. Westateaformulationwhich,
although not necessarily practical, will give us the optimal solution for a checkerboard
coloring.
Supposewearegivenanh > 0whichsatisfiesh < d. Wewanttooptimizethecheckerboard
usingthisfixedh. Thereisnowaclearchoiceofw:
(cid:26) (cid:18)1+2ehcosh(d)−e2h(cid:19)(cid:27)
w = min d, arccosh .
2
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