(4, 2)-Choosability of Planar Graphs with Forbidden Structures

Transcription

1 Mtmtcs Publctons Mtmtcs (4, )-Coosblty o Plnr Grps wt Forbn Structurs Znr Brkkyzy Iow Stt Unvrsty Crstopr Cox Crn Mllon Unvrsty Mcl Dryko Iow Stt Unvrsty, mryko@stt.u Krstn Honson Coloro Coll Mot Kumbt Unvrsty o Nv, Rno S nxt p or tonl utors Follow ts n tonl works t: ttp://lb.r.stt.u/mt_pubs Prt o t Dscrt Mtmtcs n Combntorcs Commons T complt bblorpc normton or ts tm cn b oun t ttp://lb.r.stt.u/ mt_pubs/11. For normton on ow to ct ts tm, pls vst ttp://lb.r.stt.u/ owtoct.tml. Ts Artcl s brout to you or r n opn ccss by t Mtmtcs t Iow Stt Unvrsty Dtl Rpostory. It s bn ccpt or ncluson n Mtmtcs Publctons by n utorz mnstrtor o Iow Stt Unvrsty Dtl Rpostory. For mor normton, pls contct rp@stt.u.

2 (4, )-Coosblty o Plnr Grps wt Forbn Structurs Abstrct All plnr rps r 4-colorbl n 5-coosbl, wl som plnr rps r not 4-coosbl. Dtrmnn wc proprts urnt tt plnr rp cn b color usn lsts o sz our s rcv sncnt ttnton. In trms o constrnn t structur o t rp, or ny l {,4,5,6,7}" rol="prsntton" styl="box-szn: borr-box; sply: nln; ln-t: norml; lttr-spcn: norml; wor-spcn: norml; wor-wrp: norml; wt-spc: nowrp; lot: non; rcton: ltr; mx-wt: non; mx-t: non; mn-wt: 0px; mn-t: 0px; borr: 0px; pn: 0px; mrn: 0px; poston: rltv;">l {,4,5,6,7}l {,4,5,6,7}, plnr rp s 4-coosbl t s l" rol="prsntton" styl="boxszn: borr-box; sply: nln; ln-t: norml; lttr-spcn: norml; wor-spcn: norml; worwrp: norml; wt-spc: nowrp; lot: non; rcton: ltr; mx-wt: non; mx-t: non; mnwt: 0px; mn-t: 0px; borr: 0px; pn: 0px; mrn: 0px; poston: rltv;">ll-cycl-r. In trms o constrnn t lst ssnmnt, on rnmnt o k-coosblty s coosblty wt sprton. A rp s (k, s)-coosbl t rp s colorbl rom lsts o sz k wr jcnt vrtcs v t most s common colors n tr lsts. Evry plnr rp s (4, 1)-coosbl, but tr xst plnr rps tt r not (4, )-coosbl. It s n opn quston wtr plnr rps r lwys (4, )-coosbl. A corl" rol="prsntton" styl="box-szn: borr-box; sply: nln; ln-t: norml; lttr-spcn: norml; wor-spcn: norml; wor-wrp: norml; wt-spc: nowrp; lot: non; rcton: ltr; mx-wt: non; mx-t: non; mn-wt: 0px; mn-t: 0px; borr: 0px; pn: 0px; mrn: 0px; poston: rltv;">ll-cycl s n l" rol="prsntton" styl="box-szn: borr-box; sply: nln; ln-t: norml; lttr-spcn: norml; wor-spcn: norml; wor-wrp: norml; wt-spc: nowrp; lot: non; rcton: ltr; mx-wt: non; mx-t: non; mn-wt: 0px; mn-t: 0px; borr: 0px; pn: 0px; mrn: 0px; poston: rltv;">ll-cycl wt on tonl. W monstrt or c l {5,6,7}" rol="prsntton" styl="box-szn: borr-box; sply: nln; ln-t: norml; lttr-spcn: norml; wor-spcn: norml; wor-wrp: norml; wt-spc: nowrp; lot: non; rcton: ltr; mx-wt: non; mx-t: non; mn-wt: 0px; mn-t: 0px; borr: 0px; pn: 0px; mrn: 0px; poston: rltv;">l {5,6,7}l {5,6,7} tt plnr rp s (4, )-coosbl t os not contn cor l" rol="prsntton" styl="box-szn: borr-box; sply: nln; ln-t: norml; lttr-spcn: norml; wor-spcn: norml; wor-wrp: norml; wt-spc: nowrp; lot: non; rcton: ltr; mx-wt: non; mx-t: non; mn-wt: 0px; mn-t: 0px; borr: 0px; pn: 0px; mrn: 0px; poston: rltv;">ll-cycls. Kywors Grp colorn, Plnr rp, Coosblty wt sprton, Dscrn Dscplns Dscrt Mtmtcs n Combntorcs Commnts T nl publcton s vlbl t Sprnr v ttp://x.o.or/ /s Brkkyzy, Znr, Crstopr Cox, Mcl Dryko, Krstn Honson, Mot Kumbt, Brnr Lcký, Kcy Mssrscmt t l. "(4, )-coosblty o plnr rps wt orbn structurs." Grps n Combntorcs, no. 4 (017): " Post wt Prmsson Ts rtcl s vlbl t Iow Stt Unvrsty Dtl Rpostory: ttp://lb.r.stt.u/mt_pubs/11

3 Autors Znr Brkkyzy, Crstopr Cox, Mcl Dryko, Krstn Honson, Mot Kumbt, Brnr Lcky, Kcy Mssrscmt, Kvn Moss, Ktln Nowk, Kvn F. Plmowsk, n Mcrosot Ts rtcl s vlbl t Iow Stt Unvrsty Dtl Rpostory: ttp://lb.r.stt.u/mt_pubs/11

4 (4, )-coosblty o plnr rps wt orbn structurs rxv: v1 [mt.co] 11 Dc 015 Znr Brkkyzy 1 Crstopr Cox Mcl Dryko 1 Krstn Honson 1 Mot Kumbt 1 Brnr Lcký 1, Kcy Mssrscmt 1 Kvn Moss 1 Ktln Nowk 1 Kvn F. Plmowsk 1 Drrck Stol 1,4 Dcmbr 14, 015 Abstrct All plnr rps r 4-colorbl n 5-coosbl, wl som plnr rps r not 4- coosbl. Dtrmnn wc proprts urnt tt plnr rp cn b color usn lsts o sz our s rcv sncnt ttnton. In trms o constrnn t structur o t rp, or ny l {,4,5,6,7}, plnr rp s 4-coosbl t s l-cycl-r. In trms o constrnn t lst ssnmnt, on rnmnt o k-coosblty s coosblty wt sprton. A rp s (k,s)-coosbl t rp s colorbl rom lsts o sz k wr jcnt vrtcs v t most s common colors n tr lsts. Evry plnr rp s (4,1)-coosbl, but tr xst plnr rps tt r not (4, )-coosbl. It s n opn quston wtr plnr rps r lwys (4,)-coosbl. A cor l-cycl s n l-cycl wt on tonl. W monstrt or c l {5,6,7} tt plnr rp s (4,)-coosbl t os not contn cor l-cycls. 1 Introucton A propr colorn s n ssnmnt o colors to t vrtcs o rp G suc tt jcnt vrtcs r ssn stnct colors. A (k,s)-lst ssnmnt L s uncton tt ssns lst L(v) o k colors to c vrtx v sott L(v) L(u) s wnvr uv E(G). A proprcolorn φ o G suc tt φ(v) L(v) or ll v V(G) s cll n L-colorn. W sy tt rp G s (k,s)-coosbl, or ny (k,s)-lst ssnmnt L, tr xsts n L-colorn o G. W cll ts vrton o rp colorn coosblty wt sprton. Not tt wn rp s (k, k)-coosbl, w smply sy t s k-coosbl. Obsrv tt G s (k,t)-coosbl, tn G s (k,s)-coosbl or ll s t. A notbl rsult rom Tomssn [11] stts tt vry plnr rp s 5-coosbl, so t ollows tt ll plnr rps r (5,s)-coosbl or ll s 5. Forbn crtn structurs wtn plnr rp s common rstrcton us n rp colorn. Torm 1. summrzs t currnt knowl on (, 1)-coosblty o plnr rps. Škrkovsk [1] conjctur tt ll plnr rps r (, 1)-coosbl; ts quston s stll opn n s prsnt blow s Conjctur Dprtmnt o Mtmtcs, Iow Stt Unvrsty, Ams, IA, U.S.A. {znrb,mryko,kons,mkumbt,lcky,kcymss,kmoss,knowk,kplmow,stol}@stt.u DprtmntoMtmtcl Scncs, Crn Mllon Unvrsty, Pttsbur, PA,U.S.A.cocox@nrw.cmu.u Support by NSF rnt DMS Dprtmnt o Computr Scnc, Iow Stt Unvrsty, Ams, IA, U.S.A. 1

5 Conjctur 1.1 (Škrkovsk [1]). I G s plnr rp, tn G s (,1)-coosbl. Torm 1.. A plnr rp G s (,1)-coosbl G vos ny o t ollown structurs: - -cycls (Krtocvíl, Tuz, Vot [9]). - 4-cycls (Co, Lcký, Stol [4]). - 5-cycls n 6-cycls (Co, Lcký, Stol [4]). In ts ppr, w ocus on 4-coosblty wt sprton. Krtocvíl, Tuz, n Vot [9] prov tt ll plnr rps r (4, 1)-coosbl, wl Vot [1] monstrt tt tr xst plnr rps tt r not (4, )-coosbl. It s not known ll plnr rps r (4, )-coosbl. Conjctur 1. (Krtocvíl, t l. [9]). I G s plnr rp, tn G s (4,)-coosbl. Torm 1.4 (Krtocvíl, t l. [9]). I G s plnr rp, tn G s (4,1)-coosbl. Torm 1.4 ws strntn by Krst n Lcký [8], wr t s sown tt w cn llow n npnnt st o vrtcs to v lsts o sz rtr tn 4. Torm 1.5 (KrstnLcký[8]). Lt G b plnr rp n I V(G) b n npnnt st. I L ssns lsts o colors to V(G) suc tt L(v) or vry v I, n L(v) = 4 or vry v V(G)\I, n L(u) L(v) 1 or ll uv E(G), tn G s n L-colorn. In ton to t work summrz bov, tr r svrl rsults rrn 4-coosblty. A rp s k-nrt c o ts subrps s vrtx o r t most k. Eulr s ormul mpls plnr rp wt no -cycls s -nrt n nc 4-coosbl. Ts n otr smlr rsults r lst blow n Torm 1.6. For t lst rsult n Torm 1.6, not tt cor l-cycl s n l-cycl wt n tonl connctn two o ts non-conscutv vrtcs. Torm 1.6. A plnr rp G s 4-coosbl G vos ny o t ollown structurs: - -cycls (olklor). - 4-cycls (Lm, Xu, Lu, [10]). - 5-cycls (Wn n L [14]). - 6-cycls (Fjvz, Juvn, Mor, n Škrkovsk [7]). - 7-cycls (Frz [6]). - Cor 4-cycls n cor 5-cycls (Boron n Ivnov []). Our mn rsults n ts ppr r lst blow n Torm 1.7. Not tt oubly-cor l-cycl s cor l-cycl wt n tonl. Torm 1.7. A plnr rp G s (4,)-coosbl G vos ny o t ollown structurs: - Cor 5-cycls. - Cor 6-cycls. - Cor 7-cycls. - Doubly-cor 6-cycls n oubly-cor 7-cycls. W prov c cs o Torm 1.7 sprtly. In Scton 4, w orb cor 5-cycls (s Torm 4.1). In Scton 5, w orb cor 6-cycls (s Torm 5.1); w us prts o ts proo to lso prov t cs wn orbn oubly-cor 6-cycls n oubly-cor 7-cycls (s Corollry 5.). In Scton 6, w orb cor 7-cycls (s Torm 6.). Tr r mny turs common to ll o ts proos, wc w tl n Sctons n.

6 1.1 Prlmnrs n Notton Rr to [15] or stnr rp tory trmnoloy n notton. Lt G b rp wt vrtx st V(G) n n st E(G); lt n(g) = V(G). W us K n, C n, n P n to not t complt rp, cycl rp, n pt rp, rspctvly, c on n vrtcs. T opn nboroo o vrtx, not N(v), s t st o vrtcs jcnt to v n G; t clos nboroo, not N[v], s t st N(v) {v}. T r o vrtx v, not G (v), s t numbr o vrtcs jcnt to v n G; w wrt (v) wn t rp G s clr rom t contxt. I t r o vrtx v s k, w cll v k-vrtx; t r o v s t lst k, w cll v k + -vrtx. T lnt o c, not l(), s t lnt o t c bounry wlk. I t lnt o c s k, w cll k-c; t lnt o s t lst k, w cll k + -c. Ovrvw o Mto Alloourmnrsultsustscrnmto. WrrtrrtotsurvysbyBoron[] n Crnston n Wst [5] or n ntroucton to scrn, wc s mto commonly us to obtn rsults on plnr rps. For rl numbrs v,,b, w n ntl cr vlus µ(v) = v (v) b or vry vrtx v n ν() = l() b or vry c. I v > 0, > 0 n v + = b > 0, tn Eulr s ormul mpls tt v µ(v)+ ν() = b, n t totl cr on t ntr rp s ntv. W tn n scrn ruls tt scrb mto or movn cr vlu mon vrtcs n cs wl consrvn t totl cr vlu. W monstrt tt G s mnml countrxmpl to our torm, tn vry vrtx n c ns wt nonntv cr tr t scrn procss, wc s contrcton. Intutvly, ts procss works wll wn orbn structur (suc s sort cor cycl) wt low cr. In Scton, w concrtly n rucbl conurtons. Loosly, rucbl conurton s structurc n rp G wt (4,)-lst ssnmnt L wrny L-colorn o G C xtns to n L-colorn o G. I wr looknor mnml xmplo rptt s not (4,)-coosbl, tn non o ts rucbl conurtons ppr n t rp. W n lr lst o conurtons, (C1) (C1) (s Fur ), n prov ty r rucbl usn vrous nrc constructons. T conurtons (C1) (C10) r us wn orbn cor 6- or 7-cycls, wl t conurtons (C9) (C1) r us wn orbn cor 5-cycls. T us o rnt conurtons s u to rncs n our scrn rumnts. In Scton 4, w orb cor 5-cycls n vry -c s jcnt to t most on otr - c. Morovr, -cs r not jcnt to 4-cs. Tus, our ntl cr uncton n ts cs urnts tt t only objcts wt ntv ntl cr r 4- n 5-vrtcs. In Sctons 5 n 6, w us rnt scrn strty. Our ntl cr vlus urnt tt t only objcts o ntv cr r -cs. Tus, our scrn ruls r sn to sn cr rom 5 + -cs n 4 + -vrtcs to -cs. Howvr, s w orb cor 6-cycls or cor 7-cycls, tr my b mny -cs vry clos to c otr. I G s pln rp n G s ts ul, tn lt F b t st o -cs o G n lt G b t nuc subrp o G wt vrtx st F. A clustr s mxml st o -cs tt r connct n G,.., connct componnt o G. Not tt two -cs srn n r jcnt n G, n two -cs srn only vrtx r not jcnt n G. S Fur 1 or lst o t clustrs wt mxmum cycl lnt sx n vry ntrnl vrtx o r t lst our. In ts urs, t outr cycl s not ncssrly cl cycl, ny r ll wt ry s not c, n pr o squr vrtcs rprsnt snl vrtx. Atonlly, bol s scrb sprtn

7 v v 1 v u 4 u (K) (K4) (K5) (K5b) u 1 u u u 4 v u u v u 5 u 4 1 u 1 u 1 u u 1 u 6 1 u 4 u 5 u 4 u (K5c) (K6) (K6b) (K6c) u u u 5 v 4 1 u 1 u 4 w u 4 1 u 1 5 v u 1 4 u u z 1 u (K6) (K6) (K6) (K6) w (K6) (K6) (K6j) (K6k) (K6l) (K6m) (K6n) (K6o) (K6p) (K6q) (K6r) Ts r ll o t possbl clustrs wt lonst cycl t most sx n mnmum r our. Bol s monstrt sprtn -cycls. Gry rons snt cycls tt r not cs. W roup our clustrs by t lnt o t lonst cycl n t clustr. Tus conurton (Kn) s mxmum cycl lnt o n. Fur 1: Clustrs wt mxmum cycl lnt t most sx. 4

8 -cycls, wc r cycls n pln rp wos xtror n ntror rons bot contn vrtcs not on t cycl. Ts urs r bs on t lst o clustrs us by Frz [6] n t proo tt 7-cycl-r plnr rps r 4-coosbl. For k {1,}, tr s xctly on wy to rrn k -cs n clustr. A trnl s clustr contnn xctly on -c; s (K). A mon s clustr contnn xctly two -cs; s (K4). For k, tr r multpl wys to rrn k -c n clustr. A k-n s clustr o k -cs ll ncnt to common vrtx o r t lst k +1; s (K5) n (K6b). A k-wl s clustr o k -cs ll ncnt to common vrtx o r xctly k; s (K5b) n (K6). Not tt t vrtx ncnt to ll cs o -wl s r. A k-strp s clustr o k -cs 1,..., k wr t bounrs o t -cs r sjont xcpt tt n +1 sr n or {1,...,k 1} n n + sr vrtx or {1,...,k }; s (K5) n (K6). I 1,..., k r t -cs n clustr, tn w wll prov tt t totl cr on 1,..., k tr scrn s nonntv. Tus, som o t -cs my v ntv cr, but ts s blnc by otr -cs n t clustr vn postv cr. Hnc, our proos n wt lst o ll possbl clustr typs n vryn tt c s nonntv totl cr. Wl tr r totl clustrs tt vo cor 7-cycls, w o not v tt mny css to cck. T clustrs (K5c) n (K6) (K6r) v tr bol s, monstrtn sprtn - cycl. W vo cckn ts css by usn strntn colorn sttmnt (s Torm 6.) tt llows our mnml countrxmpl to not contn ny sprtn -cycls. Rucbl Conurtons In ts scton, w scrb structurs tt cnnot ppr n mnml countrxmpl to Torm 1.7. Lt G b rp, : V(G) N, n s b nonntv ntr. A rp s -coosbl G s L-coosbl or vry lst ssnmnt L wr L(v) (v). An (,s)-lst-ssnmnt s lst ssnmnt L on G suc tt L(v) (v) or ll v V(G), L(v) L(u) s or ll s uv E(G), n L(u) L(v) = uv E(G) n (u) = (v) = 1. A rp G s (,s)-coosbl G s L-colorbl or vry (, s)-lst-ssnmnt L. Dnton.1. A conurton s trpl (C,X,x) wr C s pln rp, X V(C), n x : V(C) {0,1,, } s n xtrnl r uncton. A rp G contns t conurton (C,X,x) C pprs s n nuc subrp C o G, n or c vrtx v V(C), tr r t most x(v) s n G rom t copy o v to vrtcs not n C. For trpl (C,X,x), n t lst-sz uncton : V(C) N s { 4 x(v) v X (v) = 1 v / X. A conurton (C,X,x) s rucbl C s (,)-coosbl. Not tt rp G wt (4,)-lst ssnmnt L contns copy o rucbl conurton (C,X,x) n G X s L-coosbl, tn G s L-coosbl. Frst, w not tt (C, X, x) s rucbl conurton, tn ny wy to n btwn stnct vrtcs o X n lowr tr xtrnl r by on rsults n notr rucbl conurton. 5

9 (C1) (C) (C) (C4) (C5) (C6) (C7) (C8) (C9) (C10) (C11) (C1) (C1) (C14) (C15) (C16) (C17) (C18) (C19) (C0) (C1) In ts conurtons, s wt only on npont r xtrnl s. Vrtcs n X r ll wt wt. Fur : Rucbl conurtons. 6

10 (C1) (C) (C4) (C5) (C10) (C11) (C1) (C1) (C14) (C15) (C16) Fur : Alon-Trs Ornttons. Lmm.. Lt (C,X,x) b rucbl conurton, n suppos tt x,y X r nonjcnt vrtcs wt { x(x),x(y) 1. Lt (C,X,x ) b t conurton wr C = C +xy, X = X, n x x(v) v / {x,y} (v) = x(v) 1 v {x,y},. Tn t conurton (C,X,x ) s rucbl. Proo. Lt b t lst-sz uncton or C n not tt C s (,)-coosbl. Smlrly lt b t lst-sz uncton on t conurton (C,X,x ), n lt L b n (,)-lst ssnmnt on V(C ). Not tt (x) = (x) + 1 n (y) = (y) + 1. Lt S = L (x) L (y). I S <, tn t most on lmnt rom c o L (x) n L (y) to S untl S =. Now lt S = {,b} suc tt L (x) n b L (y), n n lst ssnmnt L on C by rmovn rom L (x) n rmovn b rom L (y). Obsrv tt L s n (,)-lst ssnmnt n nc tr xsts n L-colorn o C. Snc L(x) L(y) =, ts propr L-colorn o C s lso n L -colorn o C. W wll us Lmm. mplctly by ssumn tt C[X] pprs s n nuc subrp n our mnml countrxmpl G..1 Rucblty Proos In ts scton, w prov tt conurtons (C1) (C1) sown n Fur r rucbl..1.1 Alon-Trs Torm W wll us t clbrt Alon-Trs Torm [1] to quckly prov tt mny o our conurtons r rucbl. In ct, conurtons tt r monstrt n ts wy r rucbl or 4-coosblty, not just (4, )-coosblty. A rp D s n orntton o rp G G s t unrlyn unrct rp o D n D s no -cycls; lt + D (v) n D (v) b t out- n n-r o vrtx v n D. An Eulrn subrp o rp D s subst S E(D) suc tt, or vry vrtx v V(D), t numbr o outon s o v n S s qul to t numbr o ncomn s o v n S. Lt EE(D) b t numbr o Eulrn subrps o vn sz n EO(D) b t numbr o Eulrn subrps o o sz. 7

11 Torm. (Alon-Trs Torm [1]). Lt G b rp n : V(G) N uncton. Suppos tt tr xsts n orntton D o G suc tt + D (v) (v) 1 or vry vrtx v V(G) n EE(D) EO(D). Tn G s -coosbl. W cll n orntton n Alon-Trs orntton t stss t ypotss o Torm.. For conurton (C,X,x) n t ssoct lst-sz uncton, t sucs to monstrt n Alon-Trs orntton o C wt rspct to. S Fur or lst o Alon-Trs ornttons o svrl conurtons. Corollry.4. T ollown conurtons v Alon-Trs ornttons n nc r rucbl: (C1), (C), (C4), (C5), (C10), (C11), (C1), (C1), (C14), (C15), (C16)..1. Drct Proos In t proos blow, w consr conurton (C,X,x) wt lst-sz uncton n ssum tt n (,)-lst-ssnmnt L s vn or C. W wll monstrt tt c C s L-colorbl. Rr to Fur or rwns o t conurtons. Frst rcll t ollown ct bout lst-colorn o cycls. Fct.5. I L s -lst ssnmnt o n o cycl, tn tr os not xst n L-colorn o t cycl n only ll o t lsts r ntcl. Lmm.6. (C) s rucbl conurton. Proo. Lt v 1,...,v 4 b t vrtcs o 4-cycl wt cor v v 4 n lt v n v 4 v xtrnl r 1; t colors c(v 1 ) n c(v ) r x. Ec o v n v 4 v t lst on color n tr lsts otr tn c(v 1 ) n c(v ). Snc L(v ) or c {,4}, tr on o ts vrtcs s t lst two colors vlbl, or L(v ) L(v 4 ) = {c(v 1 ),c(v )}. In tr cs, w cn xtn t colorn. For t conurtons (C6), (C7), n (C8), lbl t vrtcs s n Fur 4: lbl t cntr vrtx v 0 n t outr vrtcs v 1,...,v 5, strtn wt t vrtx rctly bov v 0, movn clockws. v 1 v 1 v 1 v 5 v v 5 v v 5 v v 0 v 0 v 0 v 4 v v 4 v v 4 v (C6) (C7) (C8) Fur 4: Vrtx lbls or conurtons (C6), (C7), n (C8). Lmm.7. (C6) s rucbl conurton. 8

12 Proo. T colors c(v 1 ) n c(v 4 ) r trmn. I c(v 1 ) n c(v 4 ) r bot n L(v 0 ), tn slct c(v 5 ) rom L(v 5 )\(L(v 0 ) {c(v 1 ),c(v 4 )}); otrws, slct c(v 5 ) L(v 5 )\{c(v 1 ),c(v 4 )} rbtrrly. Dn L (v 0 ) = L(v 0 )\{c(v 1 ),c(v 4 ),c(v 5 )}, L (v ) = L(v )\{c(v 1 )}, n L (v ) = L(v )\{c(v 4 )} n not tt L (v ) or ll {0,,}. I L (v 0 ) = L (v ) =, tn L (v 0 ) L (v ), so t -cycl v 0 v v s n L -colorn by Fct.5. Lmm.8. (C7) s rucbl conurton. Proo. I tr xsts color L(v 1 ) L(v 4 ), strt by ssnn c(v 1 ) = c(v 4 ) = ; tn rly color t rmnn vrtcs n t ollown orr: v,v,v 0,v 5. Otrws, L(v 4 ) L(v 1 ) =. Suppos tt L(v 1 ) L(v 5 ) =. Slct color c(v 4 ) L(v 4 ). Consrn v 4 s n xtrnl vrtx n norn t v 1 v 5, t 4-cycl v 0 v 1 v v orms copy o (C4), wc s rucbl by Corollry.4. Tus, tr xsts n L-colorn o v 0,...,v 4 ; ts colorn xtns to v 5 snc L(v 1 ) L(v 5 ) =. I L(v 4 ) L(v 5 ) =, tn tr xsts n L-colorn by symmtrc rumnt. Otrws, tr xst colors L(v 1 ) \ L(v 5 ) n b L(v 4 ) \ L(v 5 ); ssn c(v 1 ) = n c(v 4 ) = b. Slct c(v ) L(v )\{c(v 1 )}. Dn L (v 0 ) = L(v 0 )\{c(v 1 ),c(v ),c(v 4 )} n L (v ) = L(v )\{c(v ),c(v 4 )}. Not tt L (v 0 ) = L (v ) = 1, tn L(v 0 ) L(v ) = {c(v ),c(v 4 )} n nc L (v 0 ) L (v ) =. Tus, t colorn xtns by rly colorn v, v 0, n v 5. Lmm.9. (C8) s rucbl conurton. Proo. I L(v 1 ) L(v ) =, tn rly color v n v ; wt rmns s (C4) n t colorn xtns. A smlr rumnt works L(v ) L(v ) =. I L(v 1 ) L(v ) =, tn L(v 1 ) L(v ) = L(v ) L(v ) = 1. Slct c(v 1 ) L(v 1 )\L(v ), c(v ) L(v )\L(v ). Dn L (v 0 ) = L(v 0 )\{c(v 1 ),c(v )}, L (v 4 ) = L(v 4 )\{c(v )}, n L (v 5 ) = L(v 5 ) \ {c(v )}. Obsrv tt w cn L -color t -cycl v 0 v 4 v 5 by Fct.5 n tn slct c(v ) L(v )\{c(v 0 )}. I tr xsts color L(v 1 ) L(v ), strt by ssnn c(v 1 ) = c(v ) = n tn ssn c(v ) L(v )\{}. DnL (v 0 ) = L(v 0 )\{,c(v )}, L (v 4 ) = L(v 4 )\{}, nl (v 5 ) = L(v 5 )\{}. Obsrv tt t -cycl v 0 v 4 v 5 s n L -colorn by Fct.5. Lmm.10. (C9) s rucbl conurton. Proo. Consr t vrtx v o rbtrry xtrnl r n lt c(v) b t color ssn to v. Lt u 1 n u b t two nbors o v n t conurton. I w rmov c(v) rom t lsts on u 1 n u, obsrv tt t lst two colors rmn n vry lst or vry vrtx o t 5-cycl. I tr s no L-colorn o t conurton, tn Fct.5 ssrts tt ll lsts v sz two n contn t sm colors; owvr, ts mpls tt L(u 1 ) = L(u ) n L(u 1 ) L(u ) =, contrcton..1. Tmplt Conurtons T conurtons (C17) (C1) r spcl css o nrl constructons cll tmplt constructons. Lt (C,X,x) b conurton wt vrtcs u,v X. A uv-pt P s cll spcl uv-pt ll ntrnl vrtcs o P v r two n C n xtrnl r two. A uv-pt P s cll n xtr-spcl uv-pt ll ntrnl vrtcs o P v xtrnl r two n r two n C, xcpt or conscutv pr xy wr x(x) = x(y) = 1, (x) = (y) =, n tr s vrtx 9

13 z / X suc tt z s common nbor to x n y, n z s not jcnt to ny otr vrtcs n C. Usn ts spcl n xtr-spcl pts, w cn scrb svrl conurtons by t ollown tmplts (s Fur 5), consstn o (B1) trnl uvw, wr x(u) = x(w) =, x(v) = 0, n xtr-spcl uv-pt P 1, n spcl vw-pt P, n (B) trnl vwr, wr x(r) =, x(w) = 1, x(v) = 0, vrtx u jcnt to v wr x(u) =, n xtr-spcl uv-pt P 1, n spcl vw-pt P. z P 1 x y u v (B1) w P P 1 x u z r y v (B) Dott lns nct spcl pts or xtr-spcl pts. Vrtcs n X r ll wt wt. Fur 5: Tmplts or rucbl conurtons. W mk som bsc obsrvtons bout spcl n xtr-spcl pts tt wll b us to prov tt ts tmplts corrspon to rucbl conurtons. Lt P b spcl uv-pt or n xtr-spcl uv-pt. For vry color L(u), lt P u () b t st contnn c color b L(v) suc tt ssnn c(u) = n c(v) = b os not xtn to n L-colorn o P. Snc w cn rly color P strtn t u untl rcn v, tr s t most on color n P u(). Furtr, u P () n only ts ry colorn procss s xctly on coc or c vrtx n P. Tus, P u() = {b} tn lso v P (b) = {}. Snc L s n (,)-lst ssnmnt, jcnt vrtcs v t most two colors n common. Tus, tr r t most two colors 1, L(u) suc tt P u( ). Morovr, obsrv tt tr r two stnct colors 1, L(u) suc tt P u( ), tn bot 1 n r n vry lst lon P n nc { 1, } L(v). I P s n xtr-spcl uv-pt wt -cycl xyz wr xy s n t pt P, tn tr color s ssn to z (s x(z) = ) tr on o x or y s tr colors vlbl or L(x) L(y) 1. Tror, P s n xtr-spcl uv-pt, tn tr s t most on color L(u) suc tt P u (). Lmm.11. All conurtons mtcn t tmplt (B1) r rucbl. Proo. Lt (C,X,x) b conurton mtcn t tmplt (B1) n lt L b n (,)-lst ssnmnt. Lt L(u) = { 1, }. Snc P 1 s n xtr-spcl pt, tr s t lst on {1,} suc tt u P 1 ( ) =. Assnc(u) =, slct c(w) L(w)\{ }nc(v) L(v)\ ( {c(u),c(w)} w P 1 (c(w)) ) ; t colorn xtns to P 1 n P. w P 10

14 Corollry.1. T conurtons (C17), (C18), n (C19) mtc t tmplt (B1), n nc ty r rucbl. Lmm.1. All conurtons mtcn t tmplt (B) r rucbl. Proo. Lt (C,X,x) b conurton mtcn t tmplt (B) n lt L b n (,)-lst ssnmnt. Lt c(r) b t unqu color n t lst L(r). Lt L(u) = { 1, }. Snc P 1 s n xtr-spcl pt, tr s t lst on {1,} suc tt u P 1 ( ) =. Assn c(u) =. I c(r) / L(v), tn slct c(w) L(w), n L(v) L(v) \ ( {c(u),c(w)} w P (c(w)) ) ; t colorn xtns to P 1 n P. I c(r) L(v), tn slct c(w) L(w) \ L(v); obsrv c(w) c(r). Tr xsts color c(v) L(v)\ ( {c(r),c(u)} w P (c(w)) ) ; t colorn xtns to P 1 n P. Corollry.14. Usn Lmm., t conurtons (C0) n (C1) mtc t tmplt (B), n nc ty r rucbl. 4 No Cor 5-Cycl In ts scton w sow t cs o orbn cor 5-cycls rom Torm 1.7. Torm 4.1. I G s pln rp not contnn cor 5-cycl, tn G s (4,)-coosbl. Proo. Lt G b countrxmpl mnmzn n(g) mon ll pln rps von cor 5- cycls wt (4,)-lst ssnmnt L suc tt G s not L-coosbl. Obsrv tt n(g) 4; n ct, δ(g) 4. Snc G s mnml countrxmpl, G os not contn ny o t rucbl conurtons (C9) (C1). I (C, X, x) s rucbl conurton, tn by Lmm. C os not ppr s subrp o G wr G (x) C (x) + x(x) or ll x V(C). Furtr, t conurtons (C1) (C1) r lr nou tt w must consr conurtons tt r orm by ntyn crtn prs o vrtcs n ts conurtons. In Appnx A, w concrtly cck ll vrtx prs tt vo crtn cor 5-cycl n n tt ll rsultn conurtons r rucbl. For c v V(G) n F(G) n ntl crs µ(v) = (v) 6 n ν() = l() 6. By Eulr s Formul, t sum o ntl crs s 1. Atr crs r ntlly ssn, t only lmnts wt ntv cr r 4-vrtcs n 5-vrtcs. Snc cor 5-cycls r orbn, tr s no -n n G n vry 4-c s jcnt to only 4 + -cs. T possbl rrnmnts o -, 4 + -, or 5 + -cs ncnt to 4- n 5-vrtcs r sown n Fur 6. Squntlly pply t ollown scrn ruls. Not tt, or vrtx v n c, w n µ (v) n ν () to b t cr on v n, rspctvly, tr pplyn rul (R). (R1) Lt v b 4-vrtx n b 4 + -c ncnt to v. I s jcnt to -c tt s lso ncnt to v, tn sns cr 1 to v; otrws, sns cr 1 to v. (R) Lt v b 5-vrtx. I s 4 + -c ncnt to v, tn sns cr 1 to v. A c s ny c ν () < 0; otrws, s non-ny. (R) I v s 5-vrtx ncnt to ny 5-c, tn v sns cr 1 to. 11

15 v v v v () (b) (c) () v v v v v () () () () () Fur 6: Possbl cyclc rrnmnts o -, 4 + -, n 5 + -cs ncnt to 4- n 5-vrtcs A vrtx v s ny vrtx µ (v) < 0; otrws, v s non-ny. (R4) I s non-ny 5 + -c ncnt to ny 5-vrtx v, tn sns cr 1 to v. W sow tt µ 4 (v) 0 or c vrtx v n ν 4 () 0 or c c. Snc t totl cr ws prsrv urn t scrn ruls, ts contrcts t ntv cr sum rom t ntl cr vlus. W bn by consrn t cr strbuton tr pplyn (R1) n (R). Lt v b vrtx. I v s 4-vrtx, tn µ(v) = n v rcvs totl cr t lst rom ts nborn cs by (R1). Furtrmor, v s not ct by ny ruls tr (R1), so µ 4 (v) 0. I v s 6 + -vrtx, tn µ(v) 0 n v s not ct by ny otr ruls, so µ 4 (v) 0. I v s 5-vrtx, tn µ(v) = 1 n v rcvs totl cr t lst 1 rom ts nborn cs by (R). Tror, or ny vrtx v, µ (v) 0. Lt b c. I s -c, tn ν() = 0 n s not ct by ny rul, so ν 4 () = 0. I s 4-c, tn ν() =. In (R1) n (R), t only cs tt sn cr 1 to snl vrtx r jcnt to -c. A 4-c jcnt to -c s cor 5-cycl, wc s orbn by ssumpton, so sns cr t most 1 to c vrtx. Snc 4-cs r not ct by ruls (R) (R4), ν 4 () 0. I s 6 + -c, tn s t lst s muc ntl cr s t s ncnt vrtcs. I v s 4-vrtx ncnt to, tn sns cr t most 1 to v by (R1) n os not sn ny cr to v by ruls (R) (R4). I v s 5-vrtx ncnt to, tn sns cr 1 to v by (R1), n possbly notr cr 1 by (R4), n os not sn cr to v by (R1) or (R). Tus sns cr t most 1 to c ncnt vrtx, n ν 4 () 0. I s 5-c, tn ν() = 4 n sns cr t most 1 to c ncnt vrtx by (R1) n (R). Obsrv tt ν () = 1, tn s ncnt to v 4-vrtcs n s jcnt to t lst on -c; ts orms (C9), contrcton. Tror, w v t ollown clm bout t structur o ny 5-vrtx. Clm 4.. I s ny 5-c, tn ν () = 1 n s jcnt to xctly on 5-vrtx. W now consr t cr strbuton tr pplyn (R). I s ny 5-c, tn ν () = 1 n s jcnt to xctly on 5-vrtx, so ν () = 0. No cs los cr n (R), tror ν () 0 or ny c. 1

16 Clm 4.. I v s ny 5-vrtx, tn v s ncnt to tr -cs, two 4 + -cs, n xctly on ny 5-c; nc µ (v) = 1. Proo. Suppos tt v s vrtx suc tt µ (v) < 0, n consr t cyclc rrnmnt o - n 4 + -cs bout v. Cs 1: v s ncnt to t lst our 4 + -cs (Furs 6() n 6()). Snc µ (v) 1 n µ (v) < 0, v s ncnt to t lst tr ny 5-cs. Hnc two o t ny 5-cs r jcnt, ormn (C1), contrcton. Cs : v s ncnt to two non-jcnt -cs n tr 4 + -cs (Fur 6()). Sncµ (v) = 1 n µ (v) < 0, v s ncnt to two ny 5-cs, 1 n. I ts two cs r jcnt, tn ty orm (C1), contrcton. Otrws, ty sr -c t s nbor n ll vrtcs ncnt to 1,, n t otr tn v r 4-vrtcs, so t vrtcs ncnt to 1 n t orm (C10), contrcton. Cs : v s ncnt to two jcnt -cs n tr 4 + -cs (Fur 6()). Snc µ (v) = 1 n µ (v) < 0, v s ncnt to two ny 5-cs, 1 n. I 1 n r jcnt tn ty orm (C1), contrcton. Tus, 1 n r not jcnt, but ty r c jcnt to -c ncnt to v. Snc s ny or c {1,}, snt cr 1 to vry 4-vrtx ncnt to. By (R1), vry 4-vrtx ncnt to s ncnt to -c jcnt to. Tror, 1 s jcnt to -c tt os not sr ny vrtcs wt t t two -cs ncnt to v, ormn on o (C0) or (C1), contrcton. Cs 4: v s ncnt to tr -cs n two 4 + -cs (Fur 6()). I v s ncnt to two ny 5-cs 1 n, tn t -c t jcnt to bot 1 n s ncnt to two 4-vrtcs, n t vrtcs ncnt to 1 n t orm (C10), contrcton. Tror, v s ncnt to xctly on ny 5-c, s clm. By (R4), vry ny 5-vrtx rcvs cr 1 rom ts unqu ncnt non-ny 5+ -c, so µ 4 (v) 0 or vry vrtx v. Ec ny 5-c s nonntv cr tr (R), so ν 4 () < 0 or som 5-c, tn sns cr by (R4), n tus s non-ny. b 1 t 1 t v 1 t v 5 v 4 v v +1 u t v +1 v +1 u t v +1 v w () A 5-c wt ν 4() < 0. (b) Clm 4.5, Cs 1. (c) Clm 4.5, Cs. Fur 7: Spcl css or 5-c wt ν 4 () < 0. 1

17 Consr t Fur 7(), wr s 5-c wt ν 4 () < 0, s ncnt to vrtcs v 1,...,v 5, v 1 s ny 5-vrtx, n 1 s t ny 5-c ncnt to v 1. Lt t 1 n t b t jcnt pr o -cs ncnt to v 1 wt t 1 jcnt to 1 n t jcnt to ; lt t b t otr -c ncnt to v 1. W mk two bsc clms bout ts rrnmnt. Clm 4.4. T vrtx v jcnt to v 1 n ncnt to t s 5 + -vrtx. Proo. I v s 4-vrtx, tn t vrtcs ncnt to 1 n t orm (C10), contrcton. Clm 4.5. I v n v +1 r conscutv vrtcs on t borr o, tn t most on o v n v +1 s ny. Proo. Suppos tt two conscutv vrtcs v n v +1 r ny 5-vrtcs. Lt n +1 b t ny 5-cs ncnt to v n v +1, rspctvly. Snc bot v n v +1 v tr ncnt -cs, s jcnt to -c t cross t v v +1. Lt u b t tr vrtx ncnt to t n consr two css. Cs 1: t s not n mon (Fur 7(b)). Snc s ny, t vrtx jcnt to u n ncnt to (wt v ) s 4-vrtx n s ncnt to -c t suc tt t s jcnt to. T vrtcs ncnt to, +1, t, n t orm on o (C15) or (C19), contrcton. Cs : t s n mon (Fur 7(c)). Lt w b t ourt vrtx n t mon n ssum, wtout loss o nrlty, tt v s jcnt to w. Lt b b t vrtx ncnt to +1 tt s not jcnt to u or v +1 lon t bounry o +1 ; snc +1 s ny, tr s -c t +1 ncnt to b n jcnt to +1. T vrtcs v n w n tos ncnt to +1 n t +1 orm on o (C17) or (C18), contrcton. By Clm 4.5, s ncnt to t most two ny vrtcs, n by Clm 4.4, v s non-ny. I s ncnt to xctly on ny 5-vrtx, tn v,v 4, n v 5 r 4-vrtcs snc µ () = 0, but tn t vrtcs ncnt to n 1 orm (C14), contrcton. Tror, s ncnt to two ny vrtcs, nsnc v s 5 + -vrtx, s ncnt toxctly two 4-vrtcs. Ec o ts rcvs cr 1, so ν 4 () = 1. By Clm 4.5, t ny vrtcs ncnt to consst o v 1 n xctly on o v or v 4. T ny 5-vrtx v otr tn v 1 s lso ncnt to tr -cs t 4,t 5, n t 6, wr t 4 n t 5 orm mon wt t 4 jcnt to. By Clm 4.4, t vrtx jcnt to v n ncnt to bot n t 6 s non-ny 5 + -vrtx. T only non-ny 5 + -vrtx ncnt to s v, n nc v s ny 5-vrtx n t 4 s ncnt to v 4. I v s 6 + -vrtx, tn ν 4 () 0. Tror, tr s unqu rrnmnt o ny vrtcs, 4-vrtcs, n 5-vrtx bout 5-c wt ν 4 () < 0 (Fur 8). For {1,}, lt b t ny 5-c ncnt to t ny 5-vrtx v. T vrtcs ncnt to, 1,, t, n t 6 orm (C16), so ts rrnmnt os not ppr wtn G; nc ν 4 () 0 or ll 5-cs. Tror, vry vrtx n c s nonntv cr tr (R4), contrctn t ntv ntl cr sum. Tus, mnml countrxmpl os not xst n vry pln rp wt no cor 5-cycl s (4, )-coosbl. 14

18 v 5 v 4 t t 4 t 1 t 5 v 1 v 1 v t t 6 Fur 8: A non-ny 5-vrtx v ncnt to non-ny 5-c wt ν 4 () < 0. 5 No Cor 6-Cycl In ts scton w sow t cs o orbn cor 6-cycls rom Torm 1.7. T cs o orbn oubly-cor 6- n 7-cycls ollows rom vry smlr rumnt. W v t ull proo or no cor 6-cycls n scrb t rncs or t proo wn w orb oublycor 6- n 7-cycls. Torm 5.1. I G s pln rp not contnn cor 6-cycl, tn G s (4,)-coosbl. Proo. Lt G b countrxmpl mnmzn n(g) mon ll pln rps von cor 6- cycls wt (4,)-lst ssnmnt L suc tt G s not L-coosbl. Obsrv tt n(g) 5; n ct, δ(g) 4. Snc G s mnml countrxmpl, G os not contn ny o t rucbl conurtons. Spcclly, w us t ct tt G vos (C) n (C4) (s Fur ). For c v V(G) n F(G) n ntl cr µ(v) = (v) 4 n ν() = l() 4. By Eulr s Formul, t ntl cr sum s 8. Snc δ(g) 4, t only lmnts o ntv cr r -cs. Snc cor 6-cycl s orbn n δ(g) 4, t clustrs (s Fur 1) r trnls (K), mons (K4), -ns (K5), 4-wls (K5b), n 4-ns wt n vrtcs nt (K5c). Spcclly not tt t 4-n (K6b) contns cor 6-cycl, so t most tr -cs n clustr sr common vrtx, unlss ty orm 4-wl (K5b) n t common vrtx s t 4-vrtx n t cntr o t wl. Apply t ollown scrn ruls, s sown n Fur 9. (R1) I s -c n s n ncnt, tn lt b t c jcnt to cross. (R1) I s 5 + -c, tn pulls cr 1 rom trou t. (R1b) I s 4-c, tn lt 1,, n b t otr s ncnt to. For c {1,,}, lt b t c jcnt to cross. For c {1,,}, t c pulls cr 1 9 rom t c trou t s n. (R) Lt v b 5 + -vrtx, n lt b n ncnt -c. (R) I v s 5-vrtx, tn v sns cr 1 (Rb) I v s 6 + -vrtx, tn v sns cr 4 9 to. to. (R) I X s clustr, tn vry -c n X s ssn t vr cr o ll -cs n X. 15

19 (R1) (R1b) (R) (Rb) v 1 v 4 9 Fur 9: Dscrn ruls n t proo o Torm 5.1. Notc tt t ruls prsrv t sum o t crs. Lt µ (v) n ν () not t cr on vrtx v or c tr rul (R). W clm tt µ (v) 0 or vry vrtx v n ν () 0 or vry c ; snc t totl cr sum s prsrv by t scrn ruls, ts contrcts t ntv cr sum rom t ntl cr vlus. Lt v b vrtx. I v s 4-vrtx, tn v s not nvolv n ny rul, so t rsultn cr s 0. I v s 6 + -vrtx, tn by (Rb) v loss cr 4 9 to c ncnt -c. Snc G vos cor 6-cycls, v s ncnt to t most 4 (v) -cs. Tus µ (v) stss µ (v) (v) (v) (v) (v) = (v) 4 0. I v s 5-vrtx, tn by (R) v loss cr 1 6-cycls, v s ncnt to t most tr -cs, so to c ncnt -c. Snc G vos cor µ (v) (v) 4 1 = (v) 5 = 0. Tror, µ (v) 0 or vry vrtx v. Lt b c. Snc 4-cs r not jcnt to 4-cs, (R1b) os not ct t cr vlu on 4-cs. Tus, ν () = 0 or vry 4-c. I s 6 + -c, tn loss cr t most 1 trou c by (R1) or (R1b), so ν () l() 4 1 l() = l() 4 0. Tror, ν () 0 or vry 6 + -c. Lt b5-c. Snc G contns no cor 6-cycls, s not jcnt to -c. Tror, loss no cr by (R1), but coul los cr usn (R1b), so ν () l() l() = 8 l() Tror, ν () 0 s5-c. Allobjctsttstrtwtnonntvcrvnonntv cr tr t scrn procss. It rmns to sow tt c clustr o -cs rcvs nou cr to rsult n nonntv cr sum. 16

20 Cs 1: (K) Lt b n solt -c. T tr jcnt cs 1,, n r ll 4 + -cs. By (R1) or (R1b), rcvs cr 1 trou c ncnt, so ν () = 1+ 1 = 0. Cs : (K4) Lt 1 n b -cs n mon clustr (K4). Tn 1 s jcnt to two 4 + -cs 1 n, n s jcnt to two 4 + -cs 1 n. By (R1) or (R1b), t clustr rcvs cr 1 trou c o t our s on t bounry o t mon. Snc ν( 1 )+ν( ) =, t cr vlu on t mon tr rul (R1) s. Snc G contns no (C), tr s 5 + -vrtx v ncnt to bot 1 n. I v s 5-vrtx, tn by (R), 1 n c rcv cr 1, n t rsultn cr on t mon s zro. I v s 6+ -vrtx, tn by (Rb), 1 n c rcv cr 4 9, n t rsultn cr on t mon s postv. Cs : (K5) Lt 1,, n b -cs n -n clustr (K5), wr s jcnt to bot 1 n. T ntl cr on ts clustr s. Tr r v s on t bounry o ts clustr, so by (R1) t clustr rcvs cr 5, rsultn n cr 4 tr (R1). Not tt t c s jcnt to bot 1 n. Snc G contns no (C), tr xsts 5 + -vrtx v ncnt to bot 1 n, n tr xsts 5 + -vrtx u ncnt to bot n. I v u, tn by (R) v sns cr t lst 1 to c o 1 n n u sns cr t lst 1 to c o n, rsultn n nonntv cr on t -n. I v = u n v s 6 + -vrtx, tn by (Rb) v sns cr 4 9 to c c 1,, n, rsultn n nonntv cr on t -n. Otrws, suppos tt v = u n v s 5-vrtx. Snc G contns no (C4), tr xsts notr 5 + -vrtx w ncnt to t lst on o 1 n. By (R) v sns cr 1 to c o 1,, n, n by (R) w sns cr t lst 1 to t lst on o 1 n, rsultn n nonntv cr on t -n. Cs 4: (K5b) Lt 1,,, n 4 b -cs n 4-wl (K5b). T ntl cr on ts clustr s 4. Trrours on tbounryo ts clustr, so by(r1) tclustr rcvs cr 4, rsultn n cr 8 tr (R1). Lt v b t 4-vrtx ncnt to ll our -cs. Lt u 1, u, u, n u 4 b t vrtcs jcnt to v, orr cyclclly suc tt vu u +1 s t bounry o t -c or {1,,} n vu 4 u 1 s t bounry o 4. Snc G contns no (C) n (v) = 4, c u s 5 + -vrtx. By (R), c u sns cr t lst to t clustr, rsultn n nonntv totl cr. Cs 5: (K5c) Lt 1,,, n 4 b -cs n 4-strp wt nt vrtcs s n (K5c). T ntl cr on ts clustr s 4. Lt v, u 1, u, u, n u 4 b t vrtcs n t 4-strp, wr v s ncnt to only 1 n 4, u 1 s ncnt to only 1 n, u s ncnt to,, n 4, u s ncnt to 1,, n, n u 4 s ncnt to only n 4. Tr r sx s on t bounry o ts clustr, so by (R1) t clustr rcvs cr 6, rsultn n cr = tr (R1). 6 Snc n orm mon, n G contns no (C), on o u n u s 5 + -vrtx. Wtout loss o nrlty, ssum u s 5 + -vrtx. Snc n 4 orm mon, n G contns no (C), on o u n u 4 s 5 + -vrtx. I u s 5 + -vrtx, tn by (R), t clustr rcvs cr t lst + rom u n u, wc rsults n nonntv totl cr. Otrws, u s 4-vrtx n u 4 s 5 + -vrtx. I u s 6 + -vrtx, tn by (R), t clustr rcvs cr t lst 4 + rom u n u 4. I u s 5-vrtx, tn snc 1 n orm mon n G contns no (C4), on o v n u 1 s 5 + -vrtx. By (R), t clustr rcvs 17

21 cr t lst + + rom u n u 4 n on o v n u 1. In tr cs, t nl cr s nonntv. W v vr tt t totl cr tr scrn s nonntv, contrctn t ntv ntl cr sum. Tus, mnml countrxmpl os not xst n vry plnr rp wt no cor 6-cycl s (4, )-coosbl. Corollry 5.. I G s pln rp not contnn oubly-cor 6-cycl or oubly-cor 7-cycl, tn G s (4, )-coosbl. Proo. Lt G b mnml countrxmpl by mnmzn n(g). Obsrv tt n(g) 4 n δ(g) 4. Snc G contns no oubly-cor 6-cycl, t clustrs r -cs (K), mons (K4), -ns (K5), 4-wls (K5b), n 4-ns wt n vrtcs nt (K5c). Us t sm scrn rumnt s n Torm 5.1, wt t ollown cns: I s 4-c, tn cn b jcnt to 4-c. Howvr, snc G contns no oublycor 7-cycl, cnnot b jcnt to -c. Tror, os not los cr by rul (R1b). I s 5-c, tn cn b jcnt to t most on -c, snc G contns no oublycor 7-cycl. By (R1) loss cr 1 cross t t srs wt, n by (R1b) loss cr t most 1 9 cross t otr our s. Tus ν () l() = 9 0. All o t otr rumnts rom t proo o Torm 5.1 ol, wc sows tt t rsultn totl cr s nonntv, n nc mnml countrxmpl os not xst. 6 No Cor 7-Cycl Torm 6.1. I G s pln rp not contnn cor 7-cycl, tn G s (4,)-coosbl. W prov t ollown strntn sttmnt: Torm 6.. Lt G b plnr rp wt no cor 7-cycl, n lt P b subrp o G, wr P s somorpc to on o P 1,P,P, or K, n ll vrtcs n V(P) r ncnt to common c. Lt L b (4,)-lst ssnmnt o G P n lt c b propr colorn o P. Tr xsts n xtnson o c to propr colorn o G suc tt c(v) L(v) or ll v V(G P). Proo. Suppos tt tr xsts countrxmpl. Slct countrxmpl (G, P, L, c) by mnmzn n(g) 1 4n(P) mon ll cor 7-cycl r pln rps, G, wt subrpp somorpc to rp n {P 1,P,P,K }, propr colorn c o P, n (4,)-lst ssnmnt L o G P suc tt c os not xtn to n L-colorn o G. W wll rr to t vrtcs o P s prcolor vrtcs. Clm 6.. G s -connct. 18

22 Proo. I G s sconnct, tn c connct componnt cn b color sprtly. Suppos tt G s cut-vrtx v. Tn tr xst connct subrps G 1 n G wr G = G 1 G n V(G 1 ) V(G ) = {v}, n(g 1 ) < n(g), n n(g ) < n(g). W cn ssum wtout loss o nrlty tt G 1 contns t lst on vrtx o P, so lt S 1 b t subrp o P contn n G 1. I G contns t lst on vrtx o P,.., v S 1, tn lt S b t subrp o P contn n G ; otrws, lt S b t vrtx v. Snc (G,P,L,c) s mnml countrxmpl, tr s n L-colorn c 1 o G 1 tt xtns t colorn on S 1. Usn t color prscrb by c 1 on v, tr xsts n L-colorn c o G tt xtns t colorn on S. T colorns c 1 n c orm n L-colorn o G, contrcton. Clm 6.4. G s no sprtn -cycls. Proo. Suppos tt P = v 1 v v s sprtn -cycl o G. Lt G 1 b t subrp o G vn by t xtror o P lon wt P, n lt G b t subrp o G vn by t ntror o P lon wt P. Snc P s sprtn, n(g 1 ) < n(g) n n(g ) < n(g). Snc t vrtcs n P sr common c, w cn ssum wtout loss o nrlty tt V(P) V(G 1 ). Snc (G,P,L,c) s mnml countrxmpl, tr xsts n L-colorn c 1 o G 1. Assn t colors rom c 1 to P. Tn tr xsts n L-colorn o G xtnn t colors on P, n totr c 1 n c orm n L-colorn o G, contrcton. Clm 6.5. I v V(P) suc tt V(P) N[v], tn t subrp o G nuc by N(v) s not somorpc to ny rp n {P 1,P,P,K }. Proo. Suppos tt tr xsts vrtx v V(P) wr ll prcolor vrtcs r n N[v] n t subrp G[N(v)] s somorpc to subrp n {P 1,P,P,K }. Tn consr t rp G = G v wt subrp P = G[N(v)]. Snc N G [v] 4, tr xsts n L-colorn c o G[N[v]]. Snc (G,P,L,c) s mnml countrxmpl, c xtns to n L-colorn o G, wc n turn xtns to n L-colorn o G, contrcton. Clm 6.6. I v V(P) s G (v), tn G (v) = n P s somorpc to P 1, P, or P. Proo. By Clm 6., G (v) 1. I G (v) = n P = K, tn G[N G (v)] s somorpc to P, contrctn Clm 6.5. Clm 6.7. P s somorpc to on o P or K. Proo. Suppos tt P s not somorpc to tr P or K. I P s somorpc to P 1, tn t vrtx o P s two conscutv nbors u 1 n u not n P; lt U = {u 1,u }. I P s somorpc to P, tn som vrtx v n P s nbor u 1 not n P tt srs c wt t n P; lt U = {u 1 }. Lt P b t subrp somorpc to P or K vn by nclun vrtcs n U. Tr xsts propr colorn c o P tt xtns t colorn on P. But tn (G,P,L,c ) s n(g) 1 4 n(p ) < n(g) 1 4n(P), so tr xsts n L-colorn o G tt xtns c, contrcton. Clm 6.8. I v V(G P), tn G (v) 4. 19

23 Proo. Suppos tt v V(G P) s r (v). Tn G v s plnr rp wt no cor 7-cycl contnn prcolor subrp P n lst ssnmnt L. Snc (G,P,L,c) s mnmum countrxmpl, G v s n L-colorn. Howvr, v s t most tr nbors n t lst our colors n t lst L(v). Tus, tr s n xtnson o t L-colorn o G v to n L-colorn o G, contrcton. Obsrv tt n(g) 4. Rcll tt n conurton (C,X,x), n L-colorn o V(C) \ X xtns to ll o C. Bcus o ts ct, G contns rucbl conurton (C,X,x), tn tr s prcolor vrtx n t st X, or ls G X s n L-colorn tt xtns to ll o G. Spcclly, w wll us t ct tt G vos (C), (C), (C4), (C5), (C6), (C7), n (C8). For c v V(G) n F(G) n µ(v) = (v) 4+δ(v) n ν() = l() 4+ε(), wr δ(v) {0,1} s vlu 1 n only v V(P), n ε() {0,1} s vlu 1 n only t bounry o s t st o prcolor vrtcs, V(P). By Eulr s Formul, t ntl cr sum s t most 1. Clms 6.6 n 6.8 ssrt tt t only ntvly-cr objcts r -cs. For vrtx v, lt t k (v) not t numbr o k-cs ncnt to v. Apply t ollown scrn ruls. Lt µ (v) n ν () not t cr on vrtx v or c tr rul (R). 8 v 1 v 4 9 (R1) (R) (Rb) (R1b) (R1c), Cs 1 (R1c), Cs Fur 10: Dscrn ruls (R1) n (R) n t proo o Torm 6.1. (R0) I v s prcolor vrtx n s n ncnt -c wt ntv cr, tn v sns cr 1 to. (R1) I s -c n s n ncnt, tn lt b t c jcnt to cross. 0

24 (R1) I s 5 + -c, tn pulls cr 8 rom trou t. (R1b) I s 4-c n s t only -c jcnt to, tn lt 1,, n b t otr s ncnt to. For c {1,,}, lt b t c jcnt to cross. For c {1,,}, t c pulls cr 1 8 rom t c trou t s n. (R1c) I s 4-c n s jcnt to two -cs 1 n (sy 1 = ), tn lt 1 n b t otr s ncnt to, wr t cs 1 n srn ts s r 6 + -cs. For c {1,}, t c pulls cr 16 rom t c trou t s n. (R) Lt v b 5 + -vrtx wt v / V(P) n lt b n ncnt -c. (R) I v s 5-vrtx, tn v sns cr 1 to, wn = mx{,t (v)}. (Rb) I v s 6 + -vrtx, tn v sns cr 1 (R) I s 6-c wt ν () < 0 n v s n ncnt 5 + -vrtx or n ncnt vrtx n V(P) wt µ 0 (v) > 0, tn v sns cr 1 4 to. W clm tt µ (v) 0 or vry vrtx v n ν () 0 or vry c. Snc t totl cr sum ws prsrv urn t scrn ruls, ts contrcts t ntv cr sum rom t ntl cr vlus. Not tt 6-cs r not ncnt to -cs snc G os not contn cor 7-cycl. Obsrv tt 6-c s ν 1 () < 0 n only ll cs jcnt to r 4-cs, n c o tos 4-cs s two jcnt -cs. Clm 6.9. Lt v b vrtx n V(P). Tn µ (v) 0. In ton, v s ncnt to 6-c wt ν 1 () < 0, tn µ 0 (v) > 0. Proo. By Clms 6.6 n 6.7, w v µ(v) = (v) 0. Not tt µ(v) 1 t (v) t 6(v), tn t nl cr µ (v) s nonntv. Snc (v) t (v) + t 6 (v), t sucs to sow tt µ 0 (v) 1 4 (v)+ 1 4 t (v). Cs 1: P = P. Lt v 1, v, n v b t vrtcs n t -pt P. For {1,,}, µ(v ) = (v ). Snc P s not somorpc to K, ts vrtcs o not orm cycl, n t c to wc ll vrtcs r ncnt s not -c. Hnc t (v ) (v ) 1. I (v ) 4, tn µ(v ) = (v ) 1 (v ) > 1 4 (v )+ 1 4 t (v ). to. I (v ) =, tn µ(v ) = 0. I =, tn v s not ncnt to ny -cs snc v 1 n v r not jcnt. I {1,} n v s jcnt to -c, tn lt v b t nbor o v not n V(P). Lt P b t subrp nuc by (V(P) {v }) \{v }, wc orms copy o P or K n G v. For ny color c(v ) L(v ) \ {c(v )}, tr xsts n L-colorn o G v s (G v,p,l,c) s not countrxmpl; ts colorn xtns to n L-colorn o G. Tus, t (v ) = 0. I v s ncnt to 6-c wt ν 1 () < 0, tn t otr c ncnt to v s 4-c tt s jcnt to two -cs. Ts rsults n cor 7-cycl, contrcton; tus (R) os not pply to v. I (v ) =, Clm 6.4 ssrts tt G s no sprtn -cycls, so tn v loss cr t most 1 n (R0). I v s ncnt to 6-c wt ν 1 () < 0, tn t otr two cs ncnt to v 1

25 r 4-cs n ts 4-cs r c jcnt to two -cs. Ts crts cor 7-cycl, contrcton, so (R) os not pply to v n µ (v ) 0. Cs : P = K. Lt v 1, v, n v bt vrtcs n t -cycl P, so µ(v ) = (v ) or c v. By Clm 6.4, G s no sprtn -cycl, so t tr vrtcs r ncnt to common -c wt ν() = 0. Tror, c vrtx v sns cr 1 to t most (v ) 1 ncnt -cs by (R0). Rcll tt (v ) by Clm 6.6. Suppos tt (v ) =. I t (v ) > 1, t subrp o G nuc by t nboroo o v s somorpc to P or K, contrctn Clm 6.5. I (v ) 4, tn µ(v ) = (v ) 1 (v ) 1 4 (v )+ 1 4 t (v ). Tror, µ (v ) 0. Tus, n ll css prcolor vrtx v s µ (v) 0. W wll now sow tt ll objcts tt strt wt nonntv cr lso n wt nonntv cr. I s 4-c, tn (R1b) n (R1c) o not pull cr rom, snc ts woul rqur to b jcnt to 4-c tt s jcnt to -c t, but tn,, n t orm oubly-cor 7-cycl. Tus, ν () = 0 or vry 4-c. I s 5-c, tn snc G contns no cor 7-cycls, s not jcnt to two -cs n s not jcnt to 4-c. Tror, loss cr t most 8 by (R1), but loss no cr usn (R1b), so ν () > 0 or vry 5-c. I s 6-c, tn s not jcnt to -c snc G contns no cor 7-cycl. Obsrv tt by Clm 6. t bounry o s smpl 6-cycl. So sns cr trou n urn (R1), t cn sn cr 1 8 trou by (R1b), or t cn sn cr 8 trou by (R1c). T only wy tt ts wll rsult n ntv cr tr (R1) n (R) s or to sn cr 8 trou c o ts sx s by (R1c); ts wll cus ν () = 6 8 = 1 4. I s prcolor vrtx v on ts bounry, tn by Clm 6.9, v s postv cr tr (R0); by (R), rcvs cr t lst 1 4, rsultn n ν () 0. I s no ncnt prcolor vrtcs, tn snc G contns no (C), som vrtx v on t bounry o s 5 + -vrtx. By (R) v sns cr 1 4 to n nc ν () 0. Obsrv t ollown clm concrnn t structur bout vrtx tt loss cr by (R). Clm Lt v b 5 + -vrtx wt t tr ncnt cs 1,, n, n cyclc orr. I v sns cr to by (R), tn 1 n r 4-cs n s 6-c. I s 7 + -c, tn by (R1) loss cr t most 8 ν () l() 4 8 l() = 5 l() 4 > 0. 8 trou c. Tus, Tror, ν () > 0 or vry 7 + -c. Nxt, w wll consr vrtx v not n V(P). I v s 4-vrtx, tn v os not los cr by ny rul, so t rsultn cr s 0. I v s 5-vrtx, lt = mx{,t (v)} n v loss cr 1 t (v) to ncnt -cs by (R). I (R) os not pply to v, tn v sns cr t most 1 to ncnt -cs n µ (v) 0. I (R) ppls to v, tn v s ncnt to cs 1,, n wr 1 n r 4-cs n s 6-c. Snc (v) = 5 n G s no cor 7-cycl, t rul (R) ppls t most onc. I (R) ppls onc, tn t (v) n v loss cr t most by (R) n cr 1 4 by (R), so µ (v) 0. I v s 6 + -vrtx, tn lt k = t (v) n l b t numbr o tms (R) ppls to v. Notc tt k 4 5(v) snc G vos cor 7-cycls. Furtr, notc tt k +l (v), snc c

26 6-c tt ns cr rom v by (R) s prc by 4-c n t cyclc orr o cs roun v. By (Rb), v cn los cr 1 to c ncnt -c, n v cn los cr t most 1 4 to c ncnt 6-c by (R). Tn v ns wt cr µ (v) (v) 4 1 k 1 4 l. I (v) = 6, tn obsrv k +l 4 n nc µ (v) 0. I (v) = 7, tn, k, n l stsy t ollown lnr prorm wt ul on vrbls 1,, n : mn 1 k 1 4 l s.t k 0 k l 0, k, l 0 mx 7 1 s.t ,, 0 40, 1 0, 1 4 ) monstrts tt 1 k 1 4l 7 40 > 4, n T ul-sbl soluton ( 1,, ) = ( tus µ (v) > 0 or vry 7 + -vrtx. It rmns to b sown tt t clustrs rcv nou cr to bcom nonntv. Snc G contns no sprtn -cycl, G os not contn t clustr (K5c) or t clustrs (K6) (K6r). Obsrv tt tr s no prcolor vrtx v o r t most tr wr ll cs ncnt to v v lnt tr. Fnlly, t s wort notn n tt G contns rucbl conurton (C,X,x), tn tr s prcolor vrtx n t st X. I vrtx v s 5 + -vrtx or v V(P), w sy v s ull; v s 6 + -vrtx or v V(P), tn v s vy. Not tt vy vrtx v sns cr 1 to c ncnt ntvly-cr -c by (R0) or (Rb). I P = K, w cll P t prcolor c. v 1 1 (K) (K4) (K5) Fur 11: Clustrs (K), (K4), n (K5) Cs 1: (K) Lt btsolt -cn(k). I stprcolorc, tnν () = ν() = 0. Otrws, tntl cron s 1.By(R1), rcvs cr 9 8 troutsbounrys, rsultn n nonntv nl cr. Cs : (K4) Lt 1 n b -cs n mon clustr (K4). Frst, suppos wtout loss o nrlty tt 1 s t prcolor c. T ntl cr o t clustr s 1. Tn rcvs cr 1 by (R0) n cr 8 by (R1), rsultn n postv nl cr. Otrws, t ntl cr on t clustr s. By (R1), 1 n rcv cr 8 trou c o t two s on t bounry o t clustr, rsultn n cr 1. I t clustr contns prcolor vrtx u, tn t rcvs cr 1 by (R0). Otrws, snc G contns no (C), tr s 5+ -vrtx v ncnt to bot 1 n. By (R), ts vrtx sns cr t lst 1 to c o t cs, rsultn n nonntv nl cr. Cs : (K5) Lt 1,, n b -cs n -n clustr (K5), wr s jcnt to bot 1 n. Suppos tt t clustr contns prcolor c, so t ntl cr on t clustr

27 s. I s prcolor, tn t clustr rcvs cr 4 1 by (R0); 1 or s prcolor, tn t clustr rcvs cr 1 by (R0) n cr 8 by (R1). In tr cs, t nl cr s nonntv. I P = K or t clustr os not contn t prcolor c, tn t ntl cr on t clustr s. By (R1), t clustr rcvs cr 5 8, rsultn n cr 9 8. Not tt t cs 1 n orm mon n t cs n orm mon. Snc G contns no (C), tr xsts ull vrtx v ncnt to bot 1 n. Smlrly, tr xsts ull vrtx u ncnt to n. I u v, tn by (R0) or (R), v sns cr t lst 1 to c o 1 n n u sns cr t lst 1 to c o n, rsultn n nonntv cr on t clustr. I u = v n v s vy vrtx, tn v sns cr 1 to c c 1,, n, rsultn n nonntv cr on t clustr. Otrws, suppos tt u = v / V(P) n v s 5-vrtx. Snc G contns no (C4), tr xsts notr ull vrtx w tt s ncnt to t lst on o 1 n. By (R), v sns cr 1 to 1,, n, n by (R0) or (R), w sns cr t lst 1 to on o 1 n, rsultn n nonntv cr on t clustr. u 1 u 1 v u 4 v u 5 u 4 u u 1 u 1 (K5b) (K6) (K6b) Fur 1: Clustrs (K5b), (K6), n (K6b) Cs 4: (K5b) Lt 1,,, n 4 b -cs n 4-wl (K5b). I t clustr contns prcolor c, tn t ntl cr on t clustr s ; t clustr rcvs cr 5 1 by (R0) n cr 8 by (R1), rsultn n postv nl cr. Otrws, t ntl cr on ts clustr s 4. By (R1), t clustr rcvs cr 4 8, rsultn n cr 5. Lt v b t 4-vrtx ncnt to ll our -cs. Lt u 1, u, u, n u 4 b t vrtcs jcnt to v, orr cyclclly suc tt vu u +1 s t bounry o t -c or {1,,} n vu 4 u 1 s t bounry o 4. Snc t clustr os not contn t prcolor c, v s not prcolor vrtx. Snc G contns no (C), c u s ull vrtx. Wn u s 5-vrtx, t s ncnt to two 7 + -cs, so u sns cr 1 to c ncnt -c by (R). Tus, c u sns cr t lst 1 to t clustr by (R0) or (R), rsultn n nonntv nl cr. Cs 5: (K6) Lt 1,,, n 4 b -cs n 4-strp clustr (K6). I t clustr contns t prcolor c, tn t ntl cr on t clustr s. I 1 or 4 s prcolor, tn t clustr rcvs cr 1 by (R0) n cr 4 8 by (R1); or s prcolor, tn t clustr rcvs cr 5 1 by (R0) n cr 5 8 by (R1). In tr cs, t rsultn nl cr s nonntv. I t clustr os not contn t prcolor c, tn t ntl cr on ts clustr s 4. By (R1), t clustr rcvs cr 6 8, rsultn n cr 7 4. Not tt or {1,,}, t cs n +1 orm mon. Snc G contns no (C), tr xsts ull vrtx v ncnt to bot n +1. Lt u 1 b ull vrtx ncnt to n. Wtout loss o nrlty, u 1 s not ncnt to 4, so tr s ull vrtx u ncnt to 1 n. I u 1 s vy vrtx, t clustr rcvs cr 1 rom u 1 by (R0) or (Rb), n cr t lst 1 rom u by (R0) or (R), rsultn n postv nl cr. Otrws, 4 u 4

28 u 1 s 5-vrtx, so u 1 sns cr 1 by (R), rsultn n cr 4. I u s ncnt to, tn u sns cr t lst 1 by (R0) or (R), rsultn n postv nl cr. Otrws, u s ncnt wt 1 n but not. I u s lr vrtx, t sns cr 1 by (R0) or (Rb). Otrws, snc G contns ntr (C) or (C4), tr s tr ull vrtx u. T clustr rcvs cr 1 rom u by (R) n cr t lst 1 rom u by (R0) or (R). In c cs, t rsultn nl cr s nonntv. Cs 6: (K6b) Lt 1,,, n 4 b -cs n 4-n clustr (K6b). Lt v b t cntr o t n, wt nbors u 1, u, u, u 4, n u 5 wr or {1,,}, n +1 r jcnt on t vu +1. I t clustr contns t prcolor c, tn t ntl cr on t clustr s. I 1 or 4 s prcolor, tn t clustr rcvs cr 4 1 by (R0) n cr 4 8 by (R1); or s prcolor, tn t clustr rcvs cr 5 1 by (R0) n cr 5 8 by (R1). In tr cs, t rsultn nl cr s postv. I t clustr os not contn t prcolor c, tn t ntl cr on ts clustr s 4. By (R1), t clustr rcvs cr 6 8, rsultn n cr 7 4. I v s vy vrtx, tn by (R0) or (Rb) v sns cr 4 1 to t clustr, rsultn n postv cr. Otrws, v / V(P) n v s 5-vrtx, so v sns cr 1 to t clustr by (R), rsultn n cr 4. I tr s vy vrtx n {u,u,u 4 }, tn tt vrtx contrbuts cr 1 to t clustr, rsultn n postv cr. I tr s no vy vrtx n {u,u,u 4 }, tn tr s t lst on 5-vrtx n {u,u,u 4 } snc G contns no (C4). I tr r multpl 5-vrtcs n {u,u,u 4 }, tn c sns cr 1 to t clustr by (R), rsultn n postv cr. I tr s only 5-vrtx w mon u,u, n u 4, tn tr s ull vrtx z {u 1,u 5 } snc G os not contn (C4) or (C5); t clustr rcvs cr 1 rom w by (R) n t lst 1 rom z by (R0) or (R), rsultn n postv nl cr. u 1 u u 1 u 6 u 4 u 5 u 4 u (K6c) v 4 1 u 1 u 4 (K6) w Fur 1: Clustrs (K6c) n (K6). Cs 7: (K6c) Lt 1,,, n 4 b t -cs o ts clustr (K6c) wr 4 s jcnt to c or {1,,}. I t clustr contns t prcolor c, tn t ntl cr on t clustr s. I on o 1, or s prcolor, t clustr rcvs cr 4 1 by (R0) n cr 4 8 by (R1). I 4 s prcolor, tn t clustr rcvs cr 6 1 by (R0). In tr cs, t rsultn nl cr s nonntv. I t clustr os not contn t prcolor c, tn t ntl cr on t clustr s 4. By (R1), t clustr rcvs cr 6 8, rsultn n cr 7 4. Lt u 1, u, u, u 4, u 5, n u 6 b t vrtcs on t bounry o t clustr orr suc tt u,u 4,u 6 r t vrtcs ncnt to 1 n, n, n n 1, rspctvly. Snc G contns no (C), tr r t lst two ull vrtcs n {u,u 4,u 6 }. By (R0) or (R), ts vrtcs c sn cr t lst 1 to t clustr, rsultn n postv totl cr. 5

29 Cs 8: (K6) Lt 1,,, n 4 b cyclclly-orr -cs n 4-wl wt cntr vrtx v wr n +1 sr common or {1,,,4}, wr ncs r tkn moulo 4; lt b -c jcnt to 4 but not ncnt to v, compltn our clustr (K6). I t clustr contns t prcolor c, tn t ntl cr on t clustr s 4. I 1 or s prcolor, tn t clustr rcvs cr 6 1 by (R0) n cr 4 8 by (R1). I s prcolor, tn t clustr rcvs cr 5 1 by (R0) n cr 4 8 by (R1). I 4 s prcolor, tn t clustr rcvs cr 7 1 by (R0) n cr 5 8 by (R1). In c o t bov css, t nl cr s nonntv. I s prcolor, tn t clustr rcvs cr 4 1 by (R0) n cr 8 by (R1), rsultn n cr 7 8. Lt N(v) = {u 1,u,u,u 4 } wr u s ncnt to n +1 or ll {1,,,4}. Snc G os not contn (C), u 1 n u r ull vrtcs. Ec o u 1 n u sns cr t lst 1 to t clustr by (R), rsultn n nonntv cr. I t clustr os not contn t prcolor c, tn t ntl cr on ts clustr s 5 n v / V(P). By (R1), t clustr rcvs cr 5 8, rsultn n cr 5 8. Snc G os not contn (C), u 1, u, u, n u 4 r ull vrtcs. By (R0) or (R), t clustr rcvs cr t lst 1 rom c o u 1 n u n cr t lst 1 rom c o u n u 4, rsultn n postv nl cr. u 4 u 5 1 u 1 5 v u 1 4 u u (K6) z 1 w (K6) u Fur 14: Clustrs (K6) n (K6). Cs 9: (K6) Lt 1,,, 4, n 5 b t cyclclly-orr -cs n 5-wl wt cntr vrtx v wr n +1 sr common or {1,,,4,5}, wr ncs r tkn moulo 5. Lt N(v) = {u 1,u,u,u 4,u 5 } wr u s ncnt to n +1 or {1,,,4,5}. I t clustr contns t prcolor c, tn t ntl cr on t clustr s 4. T clustr rcvs cr 6 1 by (R1), rsultn n postv nl cr. I t clustr os not contn t prcolor c, tn t ntl cr s 5 n v / V(P). by (R0) n cr 4 8 By (R1), t clustr rcvs cr 5 8, n by (R), t clustr rcvs cr 1 rom v, rsultn n cr Snc G os not contn (C4) or (C6), tr r t lst tr ull vrtcs n N(v). I N(v) contns t lst tr vy vrtcs, tn t clustr rcvs cr t lst 6 1 by (R0) or (Rb), rsultn n postv nl cr. I N(v) contns xctly two vy vrtcs, tn t clustr rcvs cr by (R0) or (Rb) n cr rom ull vrtx by (R), rsultn n postv cr. I N(v) contns xctly on vy vrtx, tn t clustr rcvs cr 1 1 by (R0) or (Rb) n cr rom c o two ull vrtcs by (R), rsultn n postv nl cr. I N(v) contns no vy vrtcs, tn tr r t lst tr ull vrtcs n N(v). Snc G os not contn (C4), tr r t lst two nonjcnt 5-vrtcs n N(v). Furtr, snc G os not contn (C6), (C7), or (C8), tr r t lst our 5-vrtcs n N(v). T clustr 6