A Strong Edge-Coloring of Graphs with Maximum Degree 4 Using 22 Colors

0.07 MB·English

by  Daniel Cranston

A Strong Edge-Coloring of Graphs with
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Maximum Degree 4 Using 22 Colors
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n Daniel Cranston∗
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J [e-mail]

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2 May3, 2005

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C
Abstract
.
h
In1985,Erdo˝sandNes´etrilconjecturedthatthestrongedge-coloringnumber
t
a ofagraphisbounded aboveby 5D 2 whenD isevenand 1(5D 2−2D +1)when
4 4
m D isodd. Theygaveasimpleconstructionwhichrequiresthismanycolors. The
[ conjecture hasbeen verifiedfor D ≤3. ForD =4, theconjectured bound is20.
Previously, thebest known upper bound was23 duetoHorak. Inthispaper we
1
giveanalgorithmthatusesatmost22colors.
v
3
2 1 Introduction
6
1
0 Aproperedge-coloringisanassignmentofacolortoeachedgeofagraphsothatno
6 two edges with a common endpoint receive the same color. A strong edge-coloring
0 isaproperedge-coloring,withthefurtherconditionthatnotwo edgeswiththesame
/
h colorlieonapathoflengththree. Thestrongedgechromaticnumberistheminimum
t number of colors that allow a strong edge-coloring. In this paper we consider the
a
m maximumpossiblestrongedgechromaticnumberasafunctionofthemaximumdegree
ofthegraph. Forothervariationsoftheproblem,wereferthereadertoabriefsurvey
:
v byWest[6]andapaperbyFaudree,Schelp,Gya´rfa´sandTuza[3].
Xi WeuseD todenotethemaximumdegreeofthegraph. In1985Erdo˝sandNesˇetrˇil
conjecturedthatthestrongedgechromaticnumberofagraphisatmost 5D 2forD even
r 4
a and 1(5D 2−2D +1)for D odd; they gave a constructionthat showed this numberis
4
necessary. Andersen provedthe conjecture for the case D =3 [1]. In this paper, we
considerthecaseD =4.
Erdo˝sand Nesˇetrˇil’s constructionfor D =4 is shown in figure (1a). To formthis
graph,beginwitha5-cycle,thenexpandeachvertexintotwononadjacentverticeswho
inheritallthe neighborsoftheoriginalvertex. Thegraphhas20edges, andcontains
no induced 2K (in fact, this is the largest graph that contains no induced 2K and
2 2
∗UniversityofIllinois,Urbana-Champaign.ThisworkwassupportedbytheMathematical,Information,
andComputationalSciencesDivisionsubprogramoftheOfficeofAdvancedScientificComputingResearch,
OfficeofScience,U.S.DepartmentofEnergy,underContractW-31-109-ENG-38.
1
1 INTRODUCTION 2
has D =4 [2]). Hence, in a strong edge-coloring, every edge must receive its own
color.Thebestupperboundpreviouslyknownwas23colors,provenbyHorak[4];we
improvethisupperboundto22colors.
We refer to the color classes as the integers from 1 to 22. A greedy coloring al-
gorithm sequentially colors the edges, using the least color class that is not already
prohibitedfromuseonanedgeatthetimetheedgeiscolored. Bytheneighborhood
ofanedge,wemeantheedgeswhicharedistanceatmost1fromtheedge.Intuitively,
this is the set of edges whose color could potentially restrict the color of that edge.
WeusethenotationN(e)tomeantheedgesintheneighborhoodofethatarecolored
beforeedgee. Figure(1b)showsthattheneighborhoodofanedgehassizeatmost24.
Soforeveryedgeorder,wehave|N(e)|≤24foreachedge. Thus,foreveryedgeor-
der,thegreedyalgorithmproducesastrongedge-coloringthatusesatmost25colors.
However,thereisalwayssomeorderoftheedgesforwhichthegreedyalgorithmuses
exactlytheminimumnumberofcolorsrequired. Ouraiminthispaperistoconstruct
anorderoftheedgessuchthatthegreedyalgorithmusesatmost22colors. Through-
outthis paper, when we use the term coloring, we mean strong edge-coloring. Each
connectedcomponentofGcanbecoloredindependentlyofothercomponents,sowe
assumeGisconnected. Weallowourgraphstoincludeloopsandmultipleedges. We
used todenotetheminimumdegreeofthegraphandweused(v)todenotethedegree
ofvertexv. Thegirthofagraphisthelengthoftheshortestcycle.
• •
• •
• • •
•
• •
• •
• • • • • • • e • • •
• •
• •
• •
• •
• • • •
• •
(a) (b)
Figure1:(a)Erdo˝sandNesˇetrˇil’sconstructionforD =4.Thisgraphrequires20colors.
(b)Thelargestpossibleneighborhoodofanedgehassize24.
Let v be an arbitraryvertex of a graphG. Let dist (v ) denote the distance from
v 1
vertex v to v. Let distance class i be the set of vertices at distance i from vertex v.
1
Thedistanceclassofanedgeistheminimumofthedistanceclassesofitsvertices.We
say thatanedge orderis compatiblewith vertexv if e precedese in the orderonly
1 2
when dist (e )≥dist (e ). Intuitively, we color all the edges in distance class i+1
v 1 v 2
(fartherfromv)beforewecoloranyedgeindistanceclassi(nearertov). Similarly,if
1 INTRODUCTION 3
wespecifyacycleCinthegraph,wecandefinedistanceclassitobethesetofvertices
atdistanceifromcycleC. WesayanedgeorderiscompatiblewithCife precedese
1 2
intheorderonlywhendist (e )≥dist (e ).
C 1 C 2
Lemma1 IfGisagraphwithmaximumdegree4,thenGhasastrongedge-coloring
that uses 21 colors except that it leaves uncolored those edges incident to a single
vertex. If C is a cycle in G, then G has a strong edge-coloring that uses 21 colors
exceptthatitleavesuncoloredtheedgesofC.
Proof. We first consider the case of leaving uncolored only the edges incident to a
single vertex. Let v be a vertex of G. Greedily color the edges in an order that is
compatiblewithvertexv. Supposewearecoloringedgee,notincidenttov. Letube
avertexadjacenttoanendpointofethatisonashortestpathfrometov. Thennone
ofthefouredgesincidenttouhasbeencolored,since eachedgeincidentto uisina
lowerdistanceclassthane. Thus,|N(e)|≤24−4=20.
ToprovethecaseofleavinguncoloredonlytheedgesofC,wecolortheedgesin
anordercompatiblewithC. Theargumentaboveholdsforeveryedgenotincidentto
C. Ife isincidenttoC and|C|≥4,thenatleastfouredgesintheneighborhoodofe
areedgesofC;soagain|N(e)|≤24−4=20.IfeisincidenttoCand|C|=3,thenby
countingwe see thattheneighborhoodofe hassize atmost23. Thethreeuncolored
edgesofCimplythat|N(e)|≤23−3=20.(cid:3)
Lemma1showsthatifagraphhasmaximumdegree4wecancolornearlyalledges
usingat most21 colors. In the rest of thispaper, we show thatwe can alwaysfinish
theedge-coloringusingatmostoneadditionalcolor. Theorem2isthemainresultof
thispaper. Wegivetheproofofthegeneralcase(4-regularandgirthatleastsix)now,
anddefertheothercases(whenthegraphisnot4-regularorhassmallgirth)tolemmas
4-10intheremainderofthepaper.
Theorem2 Any graph with maximum degree 4 has a strong edge-coloring with at
most22colors.
Lemma3 Any4-regulargraphwithgirthatleastsixhasastrongedge-coloringwith
atmost22colors.
Proof. By Lemma 1, we choose an arbitrary vertex v, and greedily color all edges
notincidenttov,usingatmost21colors. Nowwerecoloredgese ,e ,e ,ande (as
1 2 3 4
showninfigure2)usingcolor22. Thisallowsustogreedilyextendthecoloringtothe
fouredgesincidenttov. Edgese ,e ,e ,ande canreceivethe samecolorsince the
1 2 3 4
girthofGisatleast6. (cid:3)
Lemma 3 provesTheorem 2 for 4-regular graphs with girth at least 6. To prove
Theorem2forgraphsthatarenot4-regularandgraphswithgirthlessthansix,weuse
twoideas. InSection2,weconsidergraphsthatarenot4-regularandgraphswithgirth
atmost3. Ineach case, we exploitlocalstructureof thegraphtogivean edgeorder
with|N(e)|≤20foreveryedgeinG. InSections3and4,weconsider4-regulargraphs
withgirth4or5. Wefindpairsofedgesthatcanreceivethesamecolor. Inthiscase,
eventhough|N(e)|>21,becausenoteveryedgeinN(e)receivesadistinctcolor,we
ensurethatatmost22colorsareused.
2 GRAPHSWITHd <4ORGIRTHATMOST3 4
• • •
e
1
•
e
• 2•
• • •v • •
•e •
4
•
e
3
• • •
Figure2:Vertexvhasdegree4andthegirthofthegraphisatleast6.
2 Graphs with d < 4 or girth at most 3
Thethreelemmasinthissectionareeachprovedusingthesameidea.Wecolornearly
alledgesasinLemma1. We showthatduetothepresenceofalowdegreevertex,a
loop,adoubleedge,ora3-cycle,itispossibletoordertheremaininguncolorededges
sothatagreedycoloringusesatmost21colors.
Lemma4 Anygraphwithmaximumdegree4thathasavertexwithdegreeatmost3
hasastrongedge-coloringthatuses21colors.
Proof. We assume d(v)=3 (if actually d(v)<3, this only makes it easier to com-
pletethe coloring). Colorthe edgesin anorderthatis compatiblewith vertexv. Let
e ,e ,e betheedgesincidenttovertexv. Iftheedgesareorderede ,e ,e ,wehave
1 2 3 1 2 3
|N(e )|≤18,|N(e )|≤19and|N(e )|≤20,sotherearecolorsfore ,e ande . (cid:3)
1 2 3 1 2 3
Lemma5 A4-regulargraphwithalooporadoubleedgehasastrongedge-coloring
thatuses21colors.
Proof. Ife isa loopincidenttovertexv, we cangreedilycolortheedgesinanorder
compatiblewithvertexv(thisisverysimilartoLemma4). Sowecanassumethatthe
graphhasadoubleedge.
Letvbeoneoftheverticesincidenttothedoubleedge.Colortheedgesinanorder
thatiscompatiblewithvertexv. Lete ,e bethedoubleedgesande ,e betheother
3 4 1 2
edgesincidentto v. Then |N(e )|≤17,|N(e )|≤18,|N(e )|≤16 and |N(e )|≤17,
1 2 3 4
sotherearecolorsfore ,e ,e ,ande . (cid:3)
1 2 3 4
Lemma6 A 4-regular graph with girth 3 has a strong edge-coloring that uses 21
colors.
Proof.LetCbea3-cycleinthegraph.ByLemma1wegreedilycoloralledgesexcept
theedgesofC;thisusesatmost21colors.Anedgeofa3-cyclehasaneighborhoodof
sizeatmost20,soeachofthethreeuncolorededgessatisfies|N(e)|≤18andwecan
greedilyfinishthecoloring.(cid:3)
3 4-REGULARGRAPHSWITHGIRTHFOUR 5
3 4-regular graphs with girth four
Lemma1showsthatwecancolornearlyalledgesofthegraphusing21colors. Here
weconsider4-regulargraphsofgirthfour.Wegiveanedgeordersuchthatthegreedy
coloring uses at most 22 colors; in some cases we precolor four edges prior to the
greedycoloring.WeuseA(e)todenotethesetofcolorsavailableonedgee.
Lemma7 Any 4-regulargraphwith girth 4 has a strong edge-coloringthatuses 22
colors.
Proof. LetC bea 4-cycle,with the4edgeslabeledc (1≤i≤4)inclockwiseorder
i
andthepairofedgesnotonthe cycleandadjacentto c andc is labeleda andb
i i−1 i i
(allsubscriptsaremod4).Werefertotheedgeslabeledbya andb asincidentedges.
i i
By Lemma 1, we greedily color all edges except the edges of C and the 8 incident
edges. Thisusesatmost21colors. IftwoincidentedgesshareanendpointnotonC,
thetwoedgesformanadjacentpair. Theonlypossibilityofanadjacentpairisif a
1
orb sharesanendpointwitha orb (orsimilarlyifa orb sharesanendpointwith
1 3 3 2 2
a or b ). If the twelve uncolorededgescontain at least two adjacentpairs, then we
4 4
greedilycolortheincidentedges. Theneighborhoodofeachc hassizeatmost21,so
i
|A(c)|≥4foralli;thuswecanfinishbygreedilycoloringthefouredgesofC.
i
• •
b a
1 2
• • • •
a c b
1 1 2
c c
4 2
b c a
4 3 3
• • • •
a b
4 3
• •
Figure3:A4-cycleina4-regulargraph.
Supposetheuncolorededgescontainexactlyoneadjacentpair. Forexample,sup-
poseedgesa anda shareanendpoint. Calledgesa ,b ,a ,andb apack. Consider
2 4 1 1 3 3
thecasewhenwecanassigncolor22totwoedgesofthepack.Nowwegreedilycolor
alledgesexcepttheedgesofC. Thisusesatmost21colors(Lemma1). Eachc hasa
i
neighborhoodwithsizeatmost22. Sincecolor22isusedtwiceintheneighborhood
ofeachc,eachc satisfies|A(c)|≥4. Sowecangreedilyfinishthecoloring. Instead
i i i
considerthecasewhennopairofedgesinthepackcanreceivethesamecolor. This
impliestheexistenceofedgesbetweeneachpairofnonadjacentedgesofthepack.Call
thesefouradditionaledgesdiagonaledges. Observe(bycounting)thattheneighbor-
hoodofadiagonaledgehassizeatmost21.Sowecancolorthediagonaledgeslastin
thegreedycoloring. ThuswegreedilycoloralledgesexceptthefouredgesofC and
thefourdiagonaledges(thisusesatmost21colors). Nowwecolorthefouredgesof
4 4-REGULARGRAPHSWITHGIRTHFIVE 6
C(thefouruncoloreddiagonaledgesensurethereareenoughcolorsavailabletocolor
theedgesofC). Lastly,wecolorthefourdiagonaledges.
Finally,supposethattheuncolorededgescontainnoadjacentpairs. Inthiscasewe
will greedily color almost all edges of the graph (Lemma 1), but must do additional
workbeforehandtoensurethatafter greedilycoloringmostofthe edgeseachc will
i
satisfy|A(c)|≥4. Asabove,calledgesa ,b ,a ,andb apack. Similarly,calledges
i 1 1 3 3
a ,b ,a ,andb apack.
2 2 4 4
Considerthecasewhenwecanassigncolor21totwoedgesofonepackandassign
color22totwoedgesoftheotherpack.Wegreedilycoloralledgesbutthefouredges
ofC. Lemma1showedthatasimilargreedycoloringusedatmost21colors;however
inLemma1noneoftheedgeswereprecolored. Weadaptthatargumenttoshowthat
eveninthepresenceofthesefourprecolorededgesagreedycoloringusesatmost22
colors. Lemma1arguedtherewereatleastfouruncolorededgesintheneighborhood
of the edge being colored, so |N(e)|≤20. The same argument applies in this case
exceptthatpossiblyoneoftheedgesthatwasuncoloredinLemma1isnowcolored.
Hence|N(e)|≤21(thisfollowsfromthefactthatthefouruncolorededgesinLemma
1wereincidenttothesamevertexandinthepresentsituationatmostoneprecolored
edgeisincidenttoeachvertex).Hence,thegreedycoloringusesatmost22colors.The
neighborhoodofeachc hassizeatmost23. Sincecolors21and22areeachrepeated
i
in the neighborhoodof each c, we see that each c satisfies |A(c)|≥4. So we can
i i i
greedilyfinishthecoloring.
Instead, consider the case when we can not assign color 21 to two edges of one
pack and assign color 22 to two edges of the other pack. If no two edges in a pack
can receive the same color, this implies the existence of edges between each pair of
nonadjacentedgesofthe pack. Thesefourdiagonaledgeseachhavea neighborhood
with size at most 21. As we did above, we greedily color all edges except the four
edgesofCandthefourdiagonaledges. NowwecolorthefouredgesofC,andlastly,
wecolorthefourdiagonaledges.(cid:3)
4 4-regular graphs with girth five
Here we consider 4-regular graphs with girth five. As in the case of girth four, we
colornearlyalltheedgesbyLemma1. Intuitively,ifthereareenoughdifferentcolors
availabletobeusedontheremaininguncolorededges,weshouldbeabletocomplete
thiscoloringbygivingeachuncolorededgeitsowncolor. However,iftherearefewer
differentcolorsavailablethanthenumberofuncolorededges,thisapproachisdoomed
to fail. Hall’s Theoremformalizesthisintuition. In the languageof Hall’s Theorem,
wehavemuncolorededges,andthesetA denotesthecolorsavailabletouseonedge
i
i. ForaproofofHall’sTheorem,wereferthereadertoIntroductiontoGraphTheory
[5].
Theorem8(Hall’sTheorem) There exists a system of distinct representatives for a
familyofsets A ,A ,...,A if andonlyifthe unionofany j ofthesesets containsat
1 2 m
least jelementsforall jfrom1tom.
4 4-REGULARGRAPHSWITHGIRTHFIVE 7
We define a partialcoloringto bea strongedge-coloringexceptthatsome edges
maybe uncolored. Supposethatwe havea partialcoloring,withonlytheedgesetT
left uncolored. Let A(e) be the set of colors available to color edge e. Then Hall’s
Theoremguaranteesthatifweareunabletocompletethecoloringbygivingeachedge
itsowncolor,thereexistsasetS⊆T with|S|>|∪ A(e)|. Definethediscrepancy,
e∈S
disc(S)=|S|−|∪ A(e)|.
e∈S
Our idea is to color the set of edges with maximum discrepancy, then argue that
thiscoloringcanbeextendedtotheremaininguncolorededges.
Lemma9 LetT bethesetofuncolorededgesinapartiallycoloredgraph.LetSbea
subsetofT withmaximumdiscrepancy.ThenavalidcoloringforScanbeextendedto
avalidcoloringforallofT.
Proof. Assumetheclaimisfalse. SincethecoloringofScannotbeextendedtoT\S,
some set of edges S′ ⊆(T\S) has positive discrepancy(after coloring S). We show
thatdisc(S∪S′)>disc(S). LetR be the set ofcolorsavailable to use onat leastone
edgeof(S∪S′). LetR bethesetofcolorsavailabletouseonatleastoneedgeofS.
1
LetR bethesetofcolorsavailabletouseonatleastoneedgeofS′ aftertheedgesof
2
Shavebeencolored. Letk=disc(S). Then|S|=k+|R |and|S′|≥1+|R |. SinceS
1 2
andS′aredisjoint,weget
|S∪S′|=|S|+|S′|≥k+1+|R |+|R |>k+|R|.
1 2
ThelatterinequalityholdssinceacolorwhichisinR\R mustbeinR andthere-
1 2
forewehave|R|=|R ∪R |≤|R |+|R |. Hence
1 2 1 2
disc(S∪S′)=|S∪S′|−|R|>k=disc(S)
Thiscontradictsthemaximalityofdisc(S). Hence, anyvalidcoloringofS canbe
extendedtoavalidcoloringofT. (cid:3)
Lemma10 IfGisa4-regulargraphwithgirth5,thenGhasastrongedge-coloring
thatuses22colors.
Proof. LetC bea 5-cycle,with the5edgeslabeledc (1≤i≤5)inclockwiseorder
i
andthepairofedgesnotonthe cycleandadjacentto c andc is labeleda andb
i i−1 i i
(allsubscriptsaremod5).Werefertotheedgeslabeledbya andb asincidentedges.
i i
Edgea isatleastdistance2fromatleastoneofedgesa andb ; forif a hasedge
1 3 3 1
e to a and edge e to b then we have the 4-cycle e ,e ,b ,a . Thus (by possibly
1 3 2 3 1 2 3 3
renaminga andb )wecanassumethereisnoedgebetweenedgesa andb .
3 3 1 3
Byrepeatingthesameargument,wecanassumethereisnoedgebetweenthetwo
edgesof each of the followingpairs: (a ,b ), (a ,b ), (a ,b ), and (a ,b ). Assign
1 3 3 5 5 2 2 4
color21toedgesb andc andassigncolor22toedgesa andb . Greedilycolorall
1 3 5 2
edgesexcepttheedgesofCandtheincidentedges.Thisusesatmost22colors.
Thereare11uncolorededges;ifwecannotassignadistinctcolortoeachuncolored
edge, then Hall’s Theorem guarantees there exists a subset of the uncolored edges
with positive discrepancy. Let S be a subset of the uncolored edges with maximum
5 CONCLUSION 8
• •
a b
1 1
•
• b5 c5 c1 a2 •
• • • •
a b
5 2
c c
4 2
• •
b c a
4 3 3
• a b •
4 3
• •
Figure4:A5-cycleina4-regulargraph.
discrepancy. By countingtheuncolorededgesin the neighborhoodof eachedge, we
observe that if e is an edge of C, then |A(e)|≥8 and if e is an incident edge then
|A(e)|≥5. We can assume that S contains some edge ofC, since otherwise we can
greedilycolorS(Lemma1),thenextendthecoloringtotheremaininguncolorededges
(Lemma9). Sincedisc(S)>0and|A(e)|≥8foreachedgeofC,wehave|S|is9,10,
or11.
Suppose |S| is 9 or 10. Then since S is missing at most two uncolored edges, S
containsatleastoneofthepair(a ,b ), thepair(a ,b ), andthepair(a ,b ). Since
1 3 2 4 3 5
eachedgeinthepairsatisfies|A(e)|≥5and|∪ A(e)|≤9,somecolorisavailable
e∈S
for use on both edges of the pair. Assign the same color to both edges. Since the
neighborhood of each uncolored incident edge, e, contains at least three uncolored
edges of C, we have |N(e)|≤24−3=21; so we can greedily color the remaining
uncoloredincidentedges. NowifS containsthepair(a ,b )orthepair(a ,b )then
1 3 3 5
colorthe edgesof the 5-cyclein the orderc ,c ,c ,c ; if S containsthe pair (a ,b )
2 4 5 1 2 4
thencolortheedgesofthe5-cycleintheorderc ,c ,c ,c .
2 4 1 5
Suppose |S|is 11 and that no color is available on both edgesof any of the pairs
(a ,b ),(a ,b ),and(a ,b )(otherwisetheaboveargumentholds). Assignthesame
1 3 2 4 3 5
colorto c anda ; call itcolorx. Note thatif |A(c )|≥8, |A(a )|≥5, and |A(c )∪
1 4 1 4 1
A(a )|≤|∪ A(e)|≤10,then|A(c )∩A(a )|=6 0. Beforecolorxwasassignedtoc
4 e∈S 1 4 1
anda , ithadbeenavailableonexactlyoneedgeofeachofthethreepairs. Greedily
4
colorthosethreeedges(noneofthecolorsusedonthesethreeedgesiscolorx). Now
thethreeremaininguncoloredincidentedgeseachsatisfy|A(e)|≥3,sowecangreedily
colorthem. Greedilycolorthethreeremainingedgesintheorderc ,c ,c . (cid:3)
2 4 5
5 Conclusion
We note that it is straightforward to convert this proof to an algorithm that runs in
lineartime. We assume a data structurethatstoresallthe relevantinformationabout
eachvertex. Usingbreadth-firstsearch,wecancalculatethedistanceclasses, aswell
asimplementeachlemmainlineartime.
Anaturalquestioniswhetheritispossibletoextendtheideasofthispapertolarger
D .Thebestboundwecouldhopeforfromthetechniquesofthispaperis2D 2−3D +2.
6 ACKNOWLEDGEMENTS 9
ItisstraightforwardtoproveananalogofLemma1thatgivesastrongedge-coloring
ofGthatuses2D 2−3D +1colorsexceptthatitleavesuncoloredthoseedgesincident
toasinglevertex(however,theauthorwasunabletoproveananalogtothe“uncolored
cycle”portionofLemma1). IfGcontainsaloop,adoubleedge,oravertexofdegree
lessthanD , thenbytheanalogofLemma1Ghasastrongedge-coloringthatusesat
most2D 2−3D +1colors. UsingtheideasofLemma3, weseethatifGisD -regular
and has girth at least 6, then G has a strong edge-coloring that uses 2D 2−3D +2
colors. Thus,tocompleteaproofforgraphswithlargerD ,onemustconsiderthecase
ofregulargraphswithgirth3,4,or5.
6 Acknowledgements
This paper draws heavily on ideas from a paper by Lars Andersen [1], in which he
considers the case D (G)=3. The present author would like to thank him for that
paper. Withoutit, thisonewouldnothavebeenwritten. Theexpositionofthispaper
hasbeengreatlyimprovedbycritiquefromDavidBundeandErinChambers.
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