Theorem
Theorem2.2Foreverysimplegraph¢¡¤£¦¥¨§©.Proof.Let¥betheinputgraph.Wepresentanalgorithmthatcolorstheedgesof¥
colors.Thealgorithmhasthefollowingframework.
usingatmost
AInlpguotr:iathgmrapEhdg¥olorvinegrticeswithmaximumdegreeOutput:anedgecolouringof¥withcolours(1)let¥"!
betheemptygraphonvertices /*edgesof¥are#%$'&)(0()(1&2#'3*/4¢5to6 (2)forextendtocolouringof¥¨7¦8$tocolour¥¨75¥97¦8$A@CB#7ED Weneedtoexplainhowthegraph¥F7canbecolouredwithatmostGcolors.Inductively,suppose thatwehavecolouredtheedgesof¥F7¦8$usingatmostHIcolors.Now¥¨75¥97P8$Q@RB#7SDwhere#75£PT$&VU§. supposeCASE
1 If both vertices
T $ andU missWmoncolorX,thenwesimplycolor#7withXandwe ¥
7. obtaCinAaSEva2li.dWcoelosuripnpgofsoertthhaetgthraeprehisnocolorthatismissedbybothT$andU.LetXY$F`Missed£PT$§and X!
`Missed£U§.Since commonmissedcolour, XY$b`a we Missed£U§thereis .Otherwise,let
X an ` edge£¦TW&2£PUT Missed § § colored andX w`aiMthisXYs$e.dN£Uow§. ifTW Let £PaT%ncd&VU
U § haveabethe edgecolouredXW.
W WW TT%f Wecolours tX)h$hu&0s(0c()o(0n&isXtfr8uc$,t a”fan”(seesuchthat: Figure 2) that consists of d neighbours $e&0()(0(1& ofUanddAg different 13v3 c2noc4c1v4c2c3 noc2v2 c1 v5 c2c3 c1 c4 v1 noc1 ?
w Figure2:AfanintheexecutionofthealgorithmimplicitintheproofofVizing’stheorem. c1forallpq5&)(0(0(0&idrg,T'smissesXsand£PTYs2t$&2U§iscoloredXs;c2noneofT$&0()(0(0&Tf8$havemonmissedcolorwithU;c3forallpq5Guv&)(0(0(0&idrg,T'sdoesnotmissesX$&0()(0()&2Xs18$.Nbuottiincegtehnaetrraelqiutiirmempleienstcth3aitsp¨trgwueuoefdTgWe(sTiWndciodeesnntototmThsisasrXe$c,ojulostubreedcaXu$s&)e(0()£PT(1&2WX&2s1U8§Wt.akesthiscolourin¥¨7P8$) Therearethreepossiblesub-cases. SubCase1.thevertexTxfsatisfiesc3andTxfhasmonmissedcolorwithU.Thenwecanexpandthefan(theremustbe£¦Txft$h&2U§coloredXf).SincethedegreeofUisfinite,SubCase1mustfailat somestageandoneofthefollowingtwoothercasesshouldhappen. SubCase2.thevertexTxfhasmonmissedcolorX!
withU(seeFigure3).Thenwechangethecoloringofthefanbycoloring£¦Tyf&VU§withX!
,andcoloring£PT7&VU§withX7for45&0()(0(1&drg.Itiseasytoverifythatthisgivesavalidedgecoloringforthegraph¥
7. Wesaythatavertexinagraphmissesacolorifnoedgeincidentoniscolored.Notethateachvertexmustmissatleast onecolor.
3 13v3 noc2v2 v1 noc1 c2c1 c2noc4 c1v4c2c3 c1 v5 c3c4 c0 wnoc0 Figure3:SolutiontoSubCase2. noc3 c0c3 c013v3 noc2v2c1 c2noc4 c1v4c2noc3 c c1 v5 2c3 c4 v1 noc1 ?
wnoc0 Forbiddencases: c3 c3 c0 c0 c0 c0 c3 c3 c3c3c0 c0v3 c0v3(missesc3) Figure4:SubCase3.1with¢¡Q5aU. SubCase3.thevertexTxfmissesacolorX¤£,9©¦¥9©drg.LetX!
beacolormissedbyU.Westartfrom thevertexT£.SinceT£hasmonmissedcolorwithU,thereisanedge£¦T£h&§$§coloredwith X!
.Nowif$doesnotmissX¤£,thereis andseeifthereisanedgecoloredwith Xa!
n, edge£ andso $'&¨ on.
W §coloredwithX¤£,nowwelo©okatvertexW Bythis,weobtainedapath£whoseedges arealternativelycoloredbyX!
andX£.Thepathhasthefollowingproperties: p1thepath©£mustbefiniteandsimplesinceeachvertexofthegraph¥7P8$hasatmosttwoedgescoloredwithX!
andX£; p2thepath©£cannotbeacyclesincethevertexT£missesthecolorX¤£;andp3thevertexUisnotaninteriorvertexof©£sinceUmissesthecolorX!
(itcanonlybe¡). Let©£5BT£'&¨$Y&)(0()(0&§¡D,whereT£missescolorX¤£,¡misseseitherX¤£orX!
.If¢¡95aU,theninterchangethecolorsX!
andX£onthepathtomakevertexT£missX!
.Thencolor£PT£&2U§withX!
andcolor£PT's&VU§withXs,forpq5&0()(0(1&¥g. c3 c0c2 13v3c1 noc2v2 c2c1 noc0 c0c3 noc4 v4c2 c1 c3c4 c0v5noc3 v1 noc1 ?
w Figure5:SubCase3.2:¡5U.
4 If¡5U,wemusthave¡8$5Txf T£©tf$(seeFigure5).ThenwegrowaX!
g©Xf£path©fstarting fromwhichalsomissesTherefore,weinterchange colorcolors X¤£
X !
. andmX£uostnb©ef finiteandtomake sT%ifmmplies.sMX!
o.rTeohveenr,colorca£¦T%nfno&VUt e§nwdiathtUX !
. and£¦Ts&2U§withXsforpq5&)(0(0(0&idrg. Itisalsoeasytoseethatthisprocesscanbeimplementedbyapolynomialtimealgorithm.Weleavethedetailedimplementationofthisprocesstotheinterestedreader. Exercise.FindanexamplegraphonwhichthealgorithmimplicitinthelastproofperformstheKempe tihnotesrechadanjagceen(wttiothTc$otlhoeunrstostToWreadndassoinotne)g.ePrse,tecrhsoesnegnrainphinsc(reexaasminpgleoordfenroann-fdacetdogreizsacbolelogrreadphstsa)ratirneg
frRom colorablebutdonotseemtorequireanycolorinterchange.
5
betheemptygraphonvertices /*edgesof¥are#%$'&)(0()(1&2#'3*/4¢5to6 (2)forextendtocolouringof¥¨7¦8$tocolour¥¨75¥97¦8$A@CB#7ED Weneedtoexplainhowthegraph¥F7canbecolouredwithatmostGcolors.Inductively,suppose thatwehavecolouredtheedgesof¥F7¦8$usingatmostHIcolors.Now¥¨75¥97P8$Q@RB#7SDwhere#75£PT$&VU§. supposeCASE
1 If both vertices
T $ andU missWmoncolorX,thenwesimplycolor#7withXandwe ¥
7. obtaCinAaSEva2li.dWcoelosuripnpgofsoertthhaetgthraeprehisnocolorthatismissedbybothT$andU.LetXY$F`Missed£PT$§and X!
`Missed£U§.Since commonmissedcolour, XY$b`a we Missed£U§thereis .Otherwise,let
X an ` edge£¦TW&2£PUT Missed § § colored andX w`aiMthisXYs$e.dN£Uow§. ifTW Let £PaT%ncd&VU
U § haveabethe edgecolouredXW.
W WW TT%f Wecolours tX)h$hu&0s(0c()o(0n&isXtfr8uc$,t a”fan”(seesuchthat: Figure 2) that consists of d neighbours $e&0()(0(1& ofUanddAg different 13v3 c2noc4c1v4c2c3 noc2v2 c1 v5 c2c3 c1 c4 v1 noc1 ?
w Figure2:AfanintheexecutionofthealgorithmimplicitintheproofofVizing’stheorem. c1forallpq5&)(0(0(0&idrg,T'smissesXsand£PTYs2t$&2U§iscoloredXs;c2noneofT$&0()(0(0&Tf8$havemonmissedcolorwithU;c3forallpq5Guv&)(0(0(0&idrg,T'sdoesnotmissesX$&0()(0()&2Xs18$.Nbuottiincegtehnaetrraelqiutiirmempleienstcth3aitsp¨trgwueuoefdTgWe(sTiWndciodeesnntototmThsisasrXe$c,ojulostubreedcaXu$s&)e(0()£PT(1&2WX&2s1U8§Wt.akesthiscolourin¥¨7P8$) Therearethreepossiblesub-cases. SubCase1.thevertexTxfsatisfiesc3andTxfhasmonmissedcolorwithU.Thenwecanexpandthefan(theremustbe£¦Txft$h&2U§coloredXf).SincethedegreeofUisfinite,SubCase1mustfailat somestageandoneofthefollowingtwoothercasesshouldhappen. SubCase2.thevertexTxfhasmonmissedcolorX!
withU(seeFigure3).Thenwechangethecoloringofthefanbycoloring£¦Tyf&VU§withX!
,andcoloring£PT7&VU§withX7for45&0()(0(1&drg.Itiseasytoverifythatthisgivesavalidedgecoloringforthegraph¥
7. Wesaythatavertexinagraphmissesacolorifnoedgeincidentoniscolored.Notethateachvertexmustmissatleast onecolor.
3 13v3 noc2v2 v1 noc1 c2c1 c2noc4 c1v4c2c3 c1 v5 c3c4 c0 wnoc0 Figure3:SolutiontoSubCase2. noc3 c0c3 c013v3 noc2v2c1 c2noc4 c1v4c2noc3 c c1 v5 2c3 c4 v1 noc1 ?
wnoc0 Forbiddencases: c3 c3 c0 c0 c0 c0 c3 c3 c3c3c0 c0v3 c0v3(missesc3) Figure4:SubCase3.1with¢¡Q5aU. SubCase3.thevertexTxfmissesacolorX¤£,9©¦¥9©drg.LetX!
beacolormissedbyU.Westartfrom thevertexT£.SinceT£hasmonmissedcolorwithU,thereisanedge£¦T£h&§$§coloredwith X!
.Nowif$doesnotmissX¤£,thereis andseeifthereisanedgecoloredwith Xa!
n, edge£ andso $'&¨ on.
W §coloredwithX¤£,nowwelo©okatvertexW Bythis,weobtainedapath£whoseedges arealternativelycoloredbyX!
andX£.Thepathhasthefollowingproperties: p1thepath©£mustbefiniteandsimplesinceeachvertexofthegraph¥7P8$hasatmosttwoedgescoloredwithX!
andX£; p2thepath©£cannotbeacyclesincethevertexT£missesthecolorX¤£;andp3thevertexUisnotaninteriorvertexof©£sinceUmissesthecolorX!
(itcanonlybe¡). Let©£5BT£'&¨$Y&)(0()(0&§¡D,whereT£missescolorX¤£,¡misseseitherX¤£orX!
.If¢¡95aU,theninterchangethecolorsX!
andX£onthepathtomakevertexT£missX!
.Thencolor£PT£&2U§withX!
andcolor£PT's&VU§withXs,forpq5&0()(0(1&¥g. c3 c0c2 13v3c1 noc2v2 c2c1 noc0 c0c3 noc4 v4c2 c1 c3c4 c0v5noc3 v1 noc1 ?
w Figure5:SubCase3.2:¡5U.
4 If¡5U,wemusthave¡8$5Txf T£©tf$(seeFigure5).ThenwegrowaX!
g©Xf£path©fstarting fromwhichalsomissesTherefore,weinterchange colorcolors X¤£
X !
. andmX£uostnb©ef finiteandtomake sT%ifmmplies.sMX!
o.rTeohveenr,colorca£¦T%nfno&VUt e§nwdiathtUX !
. and£¦Ts&2U§withXsforpq5&)(0(0(0&idrg. Itisalsoeasytoseethatthisprocesscanbeimplementedbyapolynomialtimealgorithm.Weleavethedetailedimplementationofthisprocesstotheinterestedreader. Exercise.FindanexamplegraphonwhichthealgorithmimplicitinthelastproofperformstheKempe tihnotesrechadanjagceen(wttiothTc$otlhoeunrstostToWreadndassoinotne)g.ePrse,tecrhsoesnegnrainphinsc(reexaasminpgleoordfenroann-fdacetdogreizsacbolelogrreadphstsa)ratirneg
frRom colorablebutdonotseemtorequireanycolorinterchange.
5
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