(4, 2)-choosability of planar graphs with forbidden structures

Transcription

1 1 (4, )-oosility o plnr rps wit orin struturs 4 5 Znr Brikkyzy 1 Cristopr Cox Mil Diryko 1 Kirstn Honson 1 Moit Kumt 1 Brnr Liiký 1, Ky Mssrsmit 1 Kvin Moss 1 Ktln Nowk 1 Kvin F. Plmowski 1 Drrik Stol 1,4 July 7, Astrt All plnr rps r 4-olorl n 5-oosl, wil som plnr rps r not 4- oosl. Dtrminin wi proprtis urnt tt plnr rp n olor usin lists o siz our s riv siniint ttntion. In trms o onstrinin t strutur o t rp, or ny l {, 4, 5, 6, 7}, plnr rp is 4-oosl i it is l-yl-r. In trms o onstrinin t list ssinmnt, on rinmnt o k-oosility is oosility wit sprtion. A rp is (k, s)-oosl i t rp is olorl rom lists o siz k wr jnt vrtis v t most s ommon olors in tir lists. Evry plnr rp is (4, 1)-oosl, ut tr xist plnr rps tt r not (4, )-oosl. It is n opn qustion wtr plnr rps r lwys (4, )-oosl. A or l-yl is n l-yl wit on itionl. W monstrt or l {5, 6, 7} tt plnr rp is (4, )-oosl i it os not ontin or l-yls Introution A propr olorin is n ssinmnt o olors to t vrtis o rp G su tt jnt vrtis r ssin istint olors. A (k, s)-list ssinmnt L is untion tt ssins list L(v) o k olors to vrtx v so tt L(v) L(u) s wnvr uv E(G). A propr olorin φ o G su tt φ(v) L(v) or ll v V (G) is ll n L-olorin. W sy tt rp G is (k, s)-oosl i, or ny (k, s)-list ssinmnt L, tr xists n L-olorin o G. W ll tis vrition o rp olorin oosility wit sprtion. Not tt wn rp is (k, k)-oosl, w simply sy it is k-oosl. Osrv tt i G is (k, t)-oosl, tn G is (k, s)-oosl or ll s t. A notl rsult rom Tomssn [11] stts tt vry plnr rp is 5-oosl, so it ollows tt ll plnr rps r (5, s)-oosl or ll s 5. Foriin rtin struturs witin plnr rp is ommon rstrition us in rp olorin. Torm 1. summrizs t urrnt knowl on (, 1)-oosility o plnr rps. Škrkovski [1] onjtur tt ll plnr rps r (, 1)-oosl; tis qustion is still opn n is prsnt low s Conjtur Dprtmnt o Mtmtis, Iow Stt Univrsity, Ams, IA, U.S.A. {znr,miryko,kons,mkumt,liiky,kymss,kmoss,knowk,kplmow,stol}@istt.u Dprtmnt o Mtmtil Sins, Crni Mllon Univrsity, Pittsur, PA, U.S.A. oox@nrw.mu.u Support y NSF rnt DMS n DMS Dprtmnt o Computr Sin, Iow Stt Univrsity, Ams, IA, U.S.A. 1

2 Conjtur 1.1 (Škrkovski [1]). I G is plnr rp, tn G is (, 1)-oosl. Torm 1.. A plnr rp G is (, 1)-oosl i G vois ny o t ollowin struturs: - -yls (Krtovíl, Tuz, Voit [9]). - 4-yls (Coi, Liiký, Stol [4]). - 5-yls n 6-yls (Coi, Liiký, Stol [4]). In tis ppr, w ous on 4-oosility wit sprtion. Krtovíl, Tuz, n Voit [9] prov tt ll plnr rps r (4, 1)-oosl, wil Voit [1] monstrt tt tr xist plnr rps tt r not (4, )-oosl. It is not known i ll plnr rps r (4, )-oosl. Conjtur 1. (Krtovíl, t l. [9]). I G is plnr rp, tn G is (4, )-oosl. Torm 1.4 (Krtovíl, t l. [9]). I G is plnr rp, tn G is (4, 1)-oosl. Torm 1.4 ws strntn y Kirst n Liiký [8], wr it is sown tt w n llow n inpnnt st o vrtis to v lists o siz rtr tn 4. Torm 1.5 (Kirst n Liiký [8]). Lt G plnr rp n I V (G) n inpnnt st. I L ssins lists o olors to V (G) su 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-olorin. In ition to t work summriz ov, tr r svrl rsults rrin 4-oosility. A rp is k-nrt i o its surps s vrtx o r t most k. Eulr s ormul implis plnr rp wit no -yls is -nrt n n 4-oosl. Tis n otr similr rsults r list low in Torm 1.6. For t lst rsult in Torm 1.6, not tt or l-yl is n l-yl wit n itionl onntin two o its non-onsutiv vrtis. Torm 1.6. A plnr rp G is 4-oosl i G vois ny o t ollowin struturs: 5 - -yls (olklor) yls (Lm, Xu, Liu, [10]) yls (Wn n Li [14]) yls (Fijvz, Juvn, Mor, n Škrkovski [7]) yls (Frz [6]) Cor 4-yls n or 5-yls (Boroin n Ivnov []) Our min rsults in tis ppr r list low in Torm 1.7. Torm 1.7. A plnr rp G is (4, )-oosl i G vois ny o t ollowin struturs: - Cor 5-yls. - Cor 6-yls. - Cor 7-yls. W prov s o Torm 1.7 sprtly. In Stion 4, w ori or 5-yls (s Torm 4.1). In Stion 5, w ori or 6-yls (s Torm 5.1). In Stion 6, w ori or 7-yls (s Torm 6.). Tr r mny turs ommon to ll o ts proos, wi w til in Stions n.

3 Prliminris n Nottion Rr to [15] or stnr rp tory trminoloy n nottion. Lt G rp wit 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 omplt rp, yl rp, n pt rp, rsptivly, on n vrtis. T opn nioroo o vrtx, not N(v), is t st o vrtis jnt to v in G; t los nioroo, not N[v], is t st N(v) {v}. T r o vrtx v, not G (v), is t numr o vrtis jnt to v in G; w writ (v) wn t rp G is lr rom t ontxt. I t r o vrtx v is k, w ll v k-vrtx; i t r o v is t lst k (t most k), w ll v k + -vrtx (k -vrtx rsptivly). T lnt o, not l(), is t lnt o t ounry wlk. I t lnt o is k, w ll k-; i t lnt o is t lst k, w ll k + -. Ovrviw o Mto All o our min rsults us t isrin mto. W rr t rr to t survys y Boroin [] 81 n Crnston n Wst [5] or n introution to isrin, wi is mto ommonly us 8 to otin rsults on plnr rps. For rl numrs v,,, w in initil r vlus 8 µ 0 (v) = v (v) or vry vrtx v n µ 0 () = l() or vry. I v > 0, > 0, > 0, n v + =, tn Eulr s ormul implis tt v µ 0(v) + 84 µ 0() =, n 85 t totl r on t ntir rp is ntiv. W tn in isrin ruls tt sri 86 mto or movin r vlu mon vrtis n s wil onsrvin t totl r vlu. 87 W monstrt tt i G is miniml ountrxmpl to our torm, tn vry vrtx n 88 ns wit nonntiv r tr t isrin pross, wi is ontrition. Intuitivly, 89 tis pross works wll wn oriin strutur (su s sort or yl) wit low r. 90 In Stion, w onrtly in ruil oniurtions. Loosly, ruil oniurtion is 91 strutur C in rp G wit (4, )-list ssinmnt L wr ny L-olorin o G C xtns to n 9 L-olorin o G. I w r lookin or miniml xmpl o rp tt is not (4, )-oosl, tn 9 non o ts ruil oniurtions ppr in t rp. W in lr list o oniurtions, 94 (C1) (C1) (s Fiur ), n prov ty r ruil usin vrious nri onstrutions. T 95 oniurtions (C1) (C10) r us wn oriin or 6- or 7-yls, wil t oniurtions 96 (C9) (C1) r us wn oriin or 5-yls. T us o irnt oniurtions is u 97 to irns in our isrin rumnts. 98 In Stion 4, w ori or 5-yls n vry - is jnt to t most on otr Morovr, -s r not jnt to 4-s. Tus, our initil r untion in tis s 100 urnts tt t only ojts wit ntiv initil r r 4- n 5-vrtis. 101 In Stions 5 n 6, w us irnt isrin strty. Our initil r vlus urnt 10 tt t only ojts o ntiv r r -s. Tus, our isrin ruls r sin to 10 sn r rom 5 + -s n 4 + -vrtis to -s. Howvr, s w ori or 6-yls or 104 or 7-yls, tr my not mny -s vry los to otr. I G is pln rp n G is its ul, tn lt F t st o -s o G n lt G 106 t inu surp o G wit vrtx st F. A lustr is mximl st o -s tt r onnt in G, i.., onnt omponnt o G. Not tt two -s srin n r 108 jnt in G, n two -s srin only vrtx r not jnt in G. S Fiur 1 or list 109 o t lustrs wit mximum yl lnt six n vry intrnl vrtx o r t lst our. In 110 ts iurs, t outr yl is not nssrily il yl, ny r ill wit ry is not,

4 v v 1 v u 4 u (K) (K4) (K5) (K5) 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 (K5) (K6) (K6) (K6) u u u 5 v w 4 5 v u u 1 u 4 u 4 1 u 1 u u z 1 u (K6) (K6) (K6) (K6) w (K6) (K6i) (K6j) (K6k) (K6l) (K6m) (K6n) (K6o) (K6p) (K6q) (K6r) Ts r ll o t possil lustrs wit lonst yl t most six n minimum r our. Bol s monstrt sprtin -yls. Gry rions sint yls tt r not s. W roup our lustrs y t lnt o t lonst yl in t lustr. Tus oniurtion (Kni) s mximum yl lnt o n. Fiur 1: Clustrs wit mximum yl lnt t most six. 4

5 n pir o squr vrtis rprsnt sinl vrtx. Aitionlly, ol s sri sprtin -yls, wi r yls in pln rp wos xtrior n intrior rions ot ontin vrtis not on t yl. Ts iurs r s on t list o lustrs us y Frz [6] in t proo tt 7-yl-r plnr rps r 4-oosl. For k {1, }, tr is xtly on wy to rrn k -s in lustr. A trinl is lustr ontinin xtly on -; s (K). A imon is lustr ontinin xtly two -s; s (K4). For k, tr r multipl wys to rrn k il trinls in lustr. A k-n is lustr o k -s ll inint to ommon vrtx o r t lst k +1; s (K5) n (K6). A k-wl is lustr o k -s ll inint to ommon vrtx o r xtly k; s (K5) n (K6). Not tt t vrtx inint to ll s o -wl s r. A k-strip is lustr o k -s 1,..., k wr t ounris o t -s r isjoint xpt tt i n i+1 sr n or i {1,..., k 1} n i n i+ sr vrtx or i {1,..., k }; s (K5) n (K6). I 1,..., k r t -s in lustr, tn w will prov tt t totl r on 1,..., k tr isrin is nonntiv. Tus, som o t -s my v ntiv r, ut tis is ln y otr -s in t lustr vin positiv r. Hn, our proos n wit list o ll possil lustr typs n vriyin tt s nonntiv totl r. Wil tr r totl lustrs tt voi or 7-yls, w o not v tt mny ss to k. T lustrs (K5) n (K6) (K6r) v tr ol s, monstrtin sprtin - yl. W voi kin ts ss y usin strntn olorin sttmnt (s Torm 6.) tt llows our miniml ountrxmpl to not ontin ny sprtin -yls. Ruil Coniurtions In tis stion, w sri struturs tt nnot ppr in miniml ountrxmpl to Torm 1.7. Lt G rp, : V (G) N, n s nonntiv intr. A rp is -oosl i G is L-oosl or vry list ssinmnt L wr L(v) (v). An (, s)-list-ssinmnt is list ssinmnt L on G su 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) = i uv E(G) n (u) = (v) = 1. A rp G is (, s)-oosl i G is L-olorl or vry (, s)-list-ssinmnt L. Dinition.1. A oniurtion is tripl (C, X, x) wr C is pln rp, X V (C), n x : V (C) {0, 1,, } is n xtrnl r untion. A rp G ontins t oniurtion (C, X, x) i C pprs s n inu surp C o G, n or vrtx v V (C), tr r t most x(v) s in G rom t opy o v to vrtis not in C. For tripl (C, X, x), in t list-siz untion : V (C) N s { 4 x(v) v X (v) = 1 v / X. A oniurtion (C, X, x) is ruil i C is (, )-oosl. Not tt i rp G wit (4, )-list ssinmnt L ontins opy o ruil oniurtion (C, X, x) n G X is L-oosl, tn G is L-oosl. First, w not tt i (C, X, x) is ruil oniurtion, tn ny wy to n twn istint vrtis o X n lowr tir xtrnl r y on rsults in notr ruil oniurtion. 5

6 (C1) (C) (C) (C4) (C5) (C6) (C7) (C8) (C9) (C10) (C11) (C1) (C1) (C14) (C15) (C16) (C17) (C18) (C19) (C0) (C1) In ts oniurtions, s wit only on npoint r xtrnl s. Vrtis in X r ill wit wit. Fiur : Ruil oniurtions. 6

7 (C1) (C) (C4) (C5) (C10) (C11) (C1) (C1) (C14) (C15) (C16) Fiur : Alon-Trsi Orinttions Lmm.. Lt (C, X, x) ruil oniurtion, n suppos tt x, y X r nonjnt vrtis wit { x(x), x(y) 1. Lt (C, X, x ) t oniurtion wr C = C + xy, X = X, n x x(v) v / {x, y} (v) = x(v) 1 v {x, y},. Tn t oniurtion (C, X, x ) is ruil. Proo. Lt t list-siz untion or C n not tt C is (, )-oosl. Similrly lt t list-siz untion on t oniurtion (C, X, x ), n lt L n (, )-list ssinmnt 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 o L (x) n L (y) to S until S =. Now lt S = {, } su tt L (x) n L (y), n in list ssinmnt L on C y rmovin rom L (x) n rmovin rom L (y). Osrv tt L is n (, )-list ssinmnt n n tr xists n L-olorin o C. Sin L(x) L(y) =, tis propr L-olorin o C is lso n L -olorin o C. W will us Lmm. impliitly y ssumin tt C[X] pprs s n inu surp in our miniml ountrxmpl G..1 Ruiility Proos In tis stion, w prov tt oniurtions (C1) (C1) sown in Fiur r ruil..1.1 Alon-Trsi Torm 166 W will us t lrt Alon-Trsi Torm [1] to quikly prov tt mny o our oniur- 167 tions r ruil. In t, oniurtions tt r monstrt in tis wy r ruil or oosility, not just (4, )-oosility. 169 A irp D is n orinttion o rp G i G is t unrlyin unirt rp o D n D s no -yls; lt + D (v) n D (v) t out- n in-r o vrtx v in D. An Eulrin 171 surp o irp D is sust S E(D) su tt, or vry vrtx v V (D), t numr 17 o outoin s o v in S is qul to t numr o inomin s o v in S. Lt EE(D) 17 t numr o Eulrin surps o vn siz n EO(D) t numr o Eulrin surps 174 o o siz. 7

8 175 Torm. (Alon-Trsi Torm [1]). Lt G rp n : V (G) N untion. Suppos tt tr xists n orinttion D o G su tt D (v) (v) 1 or vry vrtx v V (G) n 177 EE(D) EO(D). Tn G is -oosl W ll n orinttion n Alon-Trsi orinttion i it stisis t ypotss o Torm.. For oniurtion (C, X, x) n t ssoit list-siz untion, it suis to monstrt n Alon-Trsi orinttion o C wit rspt to. S Fiur or list o Alon-Trsi orinttions o svrl oniurtions. On oul tink tt or vrtx v, t outniors r vrtis tt oul olor or v n v oul still pik olor not onlitin wit tm. I tr wr no yls in t orinttion, t orinttion woul iv n orr suitl or t ry loritm. Corollry.4. T ollowin oniurtions v Alon-Trsi orinttions n n r ruil: (C1), (C), (C4), (C5), (C10), (C11), (C1), (C1), (C14), (C15), (C16)..1. Dirt Proos In t proos low, w onsir oniurtion (C, X, x) wit list-siz untion n ssum tt n (, )-list-ssinmnt L is ivn or C. W will monstrt tt C is L-olorl. Rr to Fiur or rwins o t oniurtions. First rll t ollowin t out list-olorin o yls. Ft.5. I L is -list ssinmnt o n o yl, tn tr os not xist n L-olorin o t yl i n only i ll o t lists r intil. In t proo in tis sustion, w us sortn nottion wr or vrtx v i w not olor (v i ) y i n list L(v i ) y L i or ll i. Lmm.6. (C) is ruil oniurtion. Proo. Lt v 1,..., v 4 t vrtis o 4-yl wit or v v 4 n lt v n v 4 v xtrnl r 1; t olors 1 n r ix. E o v n v 4 v t lst on olor in tir lists otr tn 1 n. Sin L i or i {, 4}, itr on o ts vrtis s t lst two olors vill, or L L 4 = { 1, }. In itr s, w n xtn t olorin. For t oniurtions (C6), (C7), n (C8), ll t vrtis s in Fiur 4: ll t ntr vrtx v 0 n t outr vrtis v 1,..., v 5, strtin wit t vrtx irtly ov v 0, movin lokwis. 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) Fiur 4: Vrtx lls or oniurtions (C6), (C7), n (C8). 0 Lmm.7. (C6) is ruil oniurtion. 8

9 04 Proo. T olors 1 n 4 r trmin. I 1 n 4 r ot in L 0, tn slt 5 rom 05 L 5 \ (L 0 { 1, 4 }); otrwis, slt 5 L 5 \ { 1, 4 } ritrrily. Din L 0 = L 0 \ { 1, 4, 5 }, 06 L = L \ { 1 }, n L = L \ { 4 } n not tt L i or ll i {0,, }. I L 0 = L =, 07 tn L 0 L, so t -yl v 0v v s n L -olorin y Ft Lmm.8. (C7) is ruil oniurtion. Proo. I L 1 L =, tn rily olor v n v ; wt rmins is (C4) n t olorin xtns. A similr rumnt works i L L =. I L 1 L =, tn L 1 L = L L = 1. Slt 1 L 1 \ L, L \ L. Din L 0 = L 0 \ { 1, }, L 4 = L 4 \ { }, n L 5 = L 5 \ { 1 }. Osrv tt w n L -olor t -yl v 0 v 4 v 5 y Ft.5 n tn slt L \ { 0 }. I tr xists olor L 1 L, strt y ssinin 1 = = n tn ssin L \ {}. Din L 0 = L 0 \ {, }, L 4 = L 4 \ {}, n L 5 = L 5 \ {}. Osrv tt t -yl v 0 v 4 v 5 s n L -olorin y Ft.5. Lmm.9. (C8) is ruil oniurtion. 18 Proo. I tr xists olor L 1 L 4, strt y ssinin 1 = 4 = ; tn rily olor t 19 rminin vrtis in t ollowin orr: v, v, v 0, v 5. Otrwis, L 4 L 1 =. 0 Suppos tt L 1 L 5 =. Slt olor 4 L 4. Consirin v 4 s n xtrnl vrtx n 1 inorin t s v 1 v 5 n v 0 v 5, t 4-yl v 0 v 1 v v orms opy o (C4), wi is ruil y Corollry.4. Tus, tr xists n L-olorin o v 0,..., v 4 ; tis olorin xtns to v 5 sin L 1 L 5 =. I L 4 L 5 =, tn tr xists n L-olorin y symmtri rumnt. 4 Otrwis, tr xist olors L 1 \ L 5 n L 4 \ L 5 ; ssin 1 = n 4 =. Slt 5 L \ {}. Din L 0 = L 0 \ { 1,, 4 } n L = L \ {, 4 }. Not tt i L 0 = L = 1, 6 tn L 0 L = {, 4 } n n L 0 L =. Tus, t olorin xtns y rily olorin v, 7 v 0, n v Lmm.10. (C9) is ruil oniurtion. Proo. Consir t vrtx v o ritrry xtrnl r n lt (v) t olor ssin to v. Lt u 1 n u t two niors o v in t oniurtion. I w rmov (v) rom t lists on u 1 n u, osrv tt t lst two olors rmin in vry list or vry vrtx o t 5-yl. I tr is no L-olorin o t oniurtion, tn Ft.5 ssrts tt ll lists v siz two n ontin t sm olors; owvr, tis implis tt L(u 1 ) = L(u ) n L(u 1 ) L(u ) =, ontrition..1. Tmplt Coniurtions T oniurtions (C17) (C1) r spil ss o nrl onstrutions ll tmplt onstrutions. Lt (C, X, x) oniurtion wit vrtis u, v X. A uv-pt P is ll spil uv-pt i ll intrnl vrtis o P v r two in C n xtrnl r two. A uv-pt P is ll n xtr-spil uv-pt i ll intrnl vrtis v o P v xtrnl r x(v) = n r in C, not y (v), two, xpt or onsutiv pir xy wr x(x) = x(y) = 1, (x) = (y) =, n tr is vrtx z / X su tt z is ommon nior to x n y, n z is not jnt to ny otr vrtis in C. Usin ts spil n xtr-spil pts, w n sri svrl oniurtions y t ollowin tmplts (s Fiur 5), onsistin o 9

10 (B1) trinl uvw, wr x(u) = x(w) =, x(v) = 0, n xtr-spil uv-pt P 1, n spil vw-pt P, n (B) trinl vwr, wr x(r) =, x(w) = 1, x(v) = 0, vrtx u jnt to v wr x(u) =, n xtr-spil uv-pt P 1, n spil vw-pt P. z y v x u w P 1 P (B1) P 1 r u w v x y z (B) Dott lins init spil pts or xtr-spil pts. Vrtis in X r ill wit wit. Fiur 5: Tmplts or ruil oniurtions. 49 W mk som si osrvtions out spil n xtr-spil pts tt will us to 50 prov tt ts tmplts orrspon to ruil oniurtions. 51 Lt P spil uv-pt or n xtr-spil uv-pt. For vry olor L(u), lt P u () 5 t st ontinin olor L(v) su tt ssinin (u) = n (v) = os not xtn 5 to n L-olorin o P. Sin w n rily olor P strtin t u until rin v, tr is t most 54 on olor in P u (). Furtr, u P () i n only i tis ry olorin pross s xtly on 55 oi or vrtx in P. Tus, i P u P () = {}. 56 Sin L is n (, )-list ssinmnt, jnt vrtis v t most two olors in ommon. Tus, 57 tr r t most two olors 1, L(u) su tt P u ( i). Morovr, osrv tt i tr 58 r two istint olors 1, L(u) su tt P u ( i), tn ot 1 n r in vry list 59 lon P n n { 1, } L(v). 60 I P is n xtr-spil uv-pt wit -yl xyz wr xy is in t pt P, tn tr olor 61 is ssin to z (s x(z) = ) itr on o x or y s tr olors vill or L(x) L(y) 1. 6 Tror, i P is n xtr-spil uv-pt, tn tr is t most on olor L(u) su tt 6 (). u P P Lmm.11. All oniurtions mtin t tmplt (B1) r ruil. Proo. Lt (C, X, x) oniurtion mtin t tmplt (B1) n lt L n (, )-list ssinmnt. Lt L(u) = { 1, }. Sin P 1 is n xtr-spil pt, tr is t lst on i {1, } su tt P u 1 ( i ) =. Assin (u) = i, slt (w) L(w)\{ i } n (v) L(v)\ ( {(u), (w)} P w 1 ((w)) ) ; t olorin xtns to P 1 n P. Corollry.1. T oniurtions (C17), (C18), n (C19) mt t tmplt (B1), n n ty r ruil. Lmm.1. All oniurtions mtin t tmplt (B) r ruil. 10

11 Proo. Lt (C, X, x) oniurtion mtin t tmplt (B) n lt L n (, )-list ssinmnt. Lt (r) t uniqu olor in t list L(r). Lt L(u) = { 1, }. Sin P 1 is n xtr-spil pt, tr is t lst on i {1, } su tt P u 1 ( i ) =. Assin (u) = i. I (r) / L(v), tn slt (w) L(w), n L(v) L(v) \ ( {(u), (w)} P w ((w)) ) ; t olorin xtns to P 1 n P. I (r) L(v), tn slt (w) L(w) \ L(v); osrv (w) (r). Tr xists olor (v) L(v) \ ( {(r), (u)} P w ((w)) ) ; t olorin xtns to P 1 n P. Corollry.14. Usin Lmm., t oniurtions (C0) n (C1) mt t tmplt (B), n n ty r ruil. 4 No Cor 5-Cyl In tis stion w sow t s o oriin or 5-yls rom Torm 1.7. Torm 4.1. I G is pln rp not ontinin or 5-yl, tn G is (4, )-oosl. Proo. Lt G ountrxmpl minimizin n(g) mon ll pln rps voiin or 5- yls wit (4, )-list ssinmnt L su tt G is not L-oosl. Osrv tt n(g) 4; in t, δ(g) 4. Sin G is miniml ountrxmpl, G os not ontin ny o t ruil oniurtions (C9) (C1). I (C, X, x) is ruil oniurtion, tn y Lmm. C os not ppr s surp o G wr G (x) C (x) + x(x) or ll x V (C). Furtr, t oniurtions (C1) (C1) r lr nou tt w must onsir oniurtions tt r orm y intiyin rtin pirs o vrtis in ts oniurtions. In Appnix A, w onrtly k ll vrtx pirs tt voi rtin or 5-yl n in tt ll rsultin oniurtions r ruil. For v V (G) n F (G) in initil rs µ 0 (v) = (v) 6 n µ 0 µ 0 () = l() 6. By Eulr s Formul, t sum o initil rs is 1. Atr rs r initilly ssin, t only lmnts wit ntiv initil r r 4-vrtis n 5-vrtis. Sin or 5-yls r orin, tr is no -n in G n vry 4- is jnt to only 4 + -s. T possil rrnmnts o -, 4 + -, or 5 + -s inint to 4- n 5-vrtis r sown in Fiur 6. Squntilly pply t ollowin isrin ruls. Not tt, or vrtx v n, w in µ i (v) n µ i () to t r on v n, rsptivly, tr pplyin rul (Ri). (R1) Lt v 4-vrtx n inint to v. I is jnt to - tt is lso inint to v, tn sns r 1 to v; otrwis, sns r 1 to v (R) Lt v 5-vrtx. I is inint to v, tn sns r 1 A is ny i µ () < 0; otrwis, is non-ny. to v (R) I v is 5-vrtx inint to ny 5-, tn v sns r 1 A vrtx v is ny vrtx i µ (v) < 0; otrwis, v is non-ny. to. 07 (R4) I is non-ny inint to ny 5-vrtx v, tn sns r 1 to v. 11

12 v v v v () () () () v v v v v () () () () (i) Fiur 6: Possil yli rrnmnts o -, 4 + -, n 5 + -s inint to 4- n 5-vrtis 08 W sow tt µ 4 (v) 0 or vrtx v n µ 4 () 0 or. Sin t totl r 09 ws prsrv urin t isrin ruls, tis ontrits t ntiv r sum rom t initil 10 r vlus. W in y onsirin t r istriution tr pplyin (R1) n (R). 11 Lt v vrtx. I v is 4-vrtx, tn µ 0 (v) = n v rivs totl r t lst rom 1 its niorin s y (R1). Furtrmor, v is not t y ny ruls tr (R1), so µ 4 (v) 0. 1 I v is 6 + -vrtx, tn µ 0 (v) 0 n v is not t y ny otr ruls, so µ 4 (v) 0. I v is 14 5-vrtx, tn µ 0 (v) = 1 n v rivs totl r t lst 1 rom its niorin s y (R). 15 Tror, or ny vrtx v, µ (v) Lt. I is -, tn µ 0 () = 0 n is not t y ny rul, so µ 4 () = I is 4-, tn µ 0 () =. In (R1) n (R), t only s tt sn r 1 to sinl 18 vrtx r jnt to -. A 4- jnt to - is or 5-yl, wi is orin y ssumption, so sns r t most 1 to vrtx. Sin 4-s r not t y ruls 0 (R) (R4), µ 4 () 0. I is 6 + -, tn s t lst s mu initil r s it s inint 1 vrtis. I v is 4-vrtx inint to, tn sns r t most 1 to v y (R1) n os not sn ny r to v y ruls (R) (R4). I v is 5-vrtx inint to, tn sns r 1 to v y (R1), n possily notr r 1 y (R4), n os not sn r to v y (R1) or (R). 4 Tus sns r t most 1 to inint vrtx, n µ 4 () 0. 5 I is 5-, tn µ 0 () = 4 n sns r t most 1 to inint vrtx y (R1) 6 n (R). Osrv tt i µ () = 1, tn is inint to iv 4-vrtis n is jnt to t 7 lst on -; tis orms (C9), ontrition. Tror, w v t ollowin lim out t 8 strutur o ny 5-vrtx. Clim 4.. I is ny 5-, tn µ () = 1 9 n is jnt to xtly on 5-vrtx. W now onsir t r istriution tr pplyin (R). I is ny 5-, tn µ () = 1 n is jnt to xtly on 5-vrtx, so µ () = 0. No s los r in (R), tror µ () 0 or ny. Clim 4.. I v is ny 5-vrtx, tn v is inint to tr -s, two 4 + -s, n xtly on ny 5-; n µ (v) = 1. 1

13 Proo. Suppos tt v is vrtx su tt µ (v) < 0, n onsir t yli rrnmnt o - n 4 + -s out v. Cs 1: v is inint to t lst our 4 + -s (Fiurs 6() n 6()). Sin µ (v) 1 n µ (v) < 0, v is inint to t lst tr ny 5-s. Hn two o t ny 5-s r jnt, ormin (C1), ontrition. Cs : v is inint to two non-jnt -s n tr 4 + -s (Fiur 6()). Sin µ (v) = 1 n µ (v) < 0, v is inint to two ny 5-s, 1 n. I ts two s r jnt, tn ty orm (C1), ontrition. Otrwis, ty sr - t s nior n ll vrtis inint to 1,, n t otr tn v r 4-vrtis, so t vrtis inint to 1 n t orm (C10), ontrition. Cs : v is inint to two jnt -s n tr 4 + -s (Fiur 6()). Sin µ (v) = 1 n µ (v) < 0, v is inint to two ny 5-s, 1 n. I 1 n r jnt tn ty orm (C1), ontrition. Tus, 1 n r not jnt, ut ty r jnt to - inint to v. Sin i is ny or i {1, }, i snt r 1 to vry 4-vrtx inint to i. By (R1), vry 4-vrtx inint to i is inint to - jnt to i. Tror, 1 is jnt to - tt os not sr ny vrtis wit t t two -s inint to v, ormin on o (C0) or (C1), ontrition. Cs 4: v is inint to tr -s n two 4 + -s (Fiur 6(i)). I v is inint to two ny 5-s 1 n, tn t - t jnt to ot 1 n is inint to two 4-vrtis, n t vrtis inint to 1 n t orm (C10), ontrition. Tror, v is inint to xtly on ny 5-, s lim. By (R4), vry ny 5-vrtx rivs r 1 rom its uniqu inint non-ny 5+ -, so µ 4 (v) 0 or vry vrtx v. E ny 5- s nonntiv r tr (R), so i µ 4 () < 0 or som 5-, tn sns r y (R4), n tus is non-ny. 1 t 1 t v 1 t v 5 v 4 v v i+1 v i+1 u t v i i i+1 u t v i+1 v i w i () A 5- wit µ 4() < 0. () Clim 4.5, Cs 1. () Clim 4.5, Cs. Fiur 7: Spil ss or 5- wit µ 4 () < Consir t Fiur 7(), wr is 5- wit µ 4 () < 0, is inint to vrtis v 1,..., v 5, v 1 is ny 5-vrtx, n 1 is t ny 5- inint to v 1. Lt t 1 n t t jnt 1

14 pir o -s inint to v 1 wit t 1 jnt to 1 n t jnt to ; lt t t otr - inint to v 1. W mk two si lims out tis rrnmnt. Clim 4.4. T vrtx v jnt to v 1 n inint to t is 5 + -vrtx. Proo. I v is 4-vrtx, tn t vrtis inint to 1 n t orm (C10), ontrition. Clim 4.5. I v i n v i+1 r onsutiv vrtis on t orr o, tn t most on o v i n v i+1 is ny. Proo. Suppos tt two onsutiv vrtis v i n v i+1 r ny 5-vrtis. Lt i n i+1 t ny 5-s inint to v i n v i+1, rsptivly. Sin ot v i n v i+1 v tr inint -s, is jnt to - t ross t v i v i+1. Lt u t tir vrtx inint to t n onsir two ss. Cs 1: t is not in imon (Fiur 7()). Sin i is ny, t vrtx jnt to u n inint to i (wit v i ) is 4-vrtx n is inint to - t i su tt t i is jnt to i. T vrtis inint to i, i+1, t, n t i orm on o (C15) or (C19), ontrition. Cs : t is in imon (Fiur 7()). Lt w t ourt vrtx in t imon n ssum, witout loss o nrlity, tt v i is jnt to w. Lt t vrtx inint to i+1 tt is not jnt to u or v i+1 lon t ounry o i+1 ; sin i+1 is ny, tr is - t i+1 inint to n jnt to i+1. T vrtis v i n w n tos inint to i+1 n t i+1 orm on o (C17) or (C18), ontrition By Clim 4.5, is inint to t most two ny vrtis, n y Clim 4.4, v is non-ny. I 80 is inint to xtly on ny 5-vrtx, tn v, v 4, n v 5 r 4-vrtis sin µ () = 0, ut 81 tn t vrtis inint to n 1 orm (C14), ontrition. 8 Tror, is inint to two ny vrtis, n sin v is 5 + -vrtx y Clim 4.4, is inint to xtly two 4-vrtis. E o ts rivs r 1, so µ 4 () = 1. By Clim 4.5, 84 t ny vrtis inint to onsist o v 1 n xtly on o v or v 4. T ny 5-vrtx v i 85 otr tn v 1 is lso inint to tr -s t 4, t 5, n t 6, wr t 4 n t 5 orm imon wit t 4 86 jnt to. By Clim 4.4, t vrtx jnt to v i n inint to ot n t 6 is non-ny vrtx. T only non-ny 5 + -vrtx inint to is v, n n v is ny 5-vrtx n 88 t 4 is inint to v 4. I v is 6 + -vrtx, tn µ 4 () 0. Tror, tr is uniqu rrnmnt o 89 ny vrtis, 4-vrtis, n 5-vrtx out 5- wit µ 4 () < 0 (Fiur 8). For i {1, }, 90 lt i t ny 5- inint to t ny 5-vrtx v i. 91 T vrtis inint to, 1,, t, n t 6 orm (C16), so tis rrnmnt os not ppr 9 witin G; n µ 4 () 0 or ll 5-s. Tror, vry vrtx n s nonntiv 9 r tr (R4), ontritin t ntiv initil r sum. Tus, miniml ountrxmpl 94 os not xist n vry pln rp wit no or 5-yl is (4, )-oosl No Cor 6-Cyl In tis stion w sow t s o oriin or 6-yls rom Torm 1.7. Torm 5.1. I G is pln rp not ontinin ny or 6-yl, tn G is (4, )-oosl. 14

15 v 5 v 4 t t 4 t 1 t 5 v 1 v 1 v t t 6 Fiur 8: A non-ny 5-vrtx v inint to non-ny 5- wit µ 4 () < W prov t ollowin strntn sttmnt. Torm 5.. Lt G pln rp wit no or 6-yl, n lt P surp o G, wr P is isomorpi to on o P 1, P, P, or K, n ll vrtis in V (P ) r inint to ommon. Lt L (4, )-list ssinmnt o G P n lt propr olorin o P. Tr xists n xtnsion o to propr olorin o G su tt (v) L(v) or ll v V (G P ). 40 Proo. Suppos tt tr xists ountrxmpl. Slt ountrxmpl (G, P, L, ) y minimizin n(g) 1 4n(P ) n sujt to tt y minimizin t numr o s mon ll or yl r pln rps, G, wit surp P isomorpi to rp in {P 1, P, P, K }, propr 406 olorin o P, n (4, )-list ssinmnt L o G P su tt os not xtn to n L-olorin 407 o G. W will rr to t vrtis o P s prolor vrtis. 408 Clim 5.. G is -onnt Proo. I G is isonnt, tn onnt omponnt n olor sprtly y t minimlity o G. Suppos tt G s ut-vrtx v. Tn tr xist onnt surps 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 n ssum witout loss o nrlity tt G 1 ontins t lst on vrtx o P, so lt S 1 t surp o P ontin in G 1. Lt S = {v} (V (G ) V (P )). Sin (G, P, L, ) is miniml ountrxmpl, tr is n L-olorin 1 o G 1 tt xtns t olorin on S 1. Usin t olor prsri y 1 on v, tr xists n L-olorin o G tt xtns t olorin on S. T olorins 1 n orm n L-olorin o G, ontrition. Clim 5.4. G s no sprtin -yls. Proo. Suppos tt P = v 1 v v is sprtin -yl o G. Lt G 1 t surp o G ivn y t xtrior o P lon wit P, n lt G t surp o G ivn y t intrior o P lon wit P. Sin P is sprtin, n(g 1 ) < n(g) n n(g ) < n(g). Sin t vrtis in P sr ommon, w n ssum witout loss o nrlity tt V (P ) V (G 1 ). Sin (G, P, L, ) is miniml ountrxmpl, tr xists n L-olorin 1 o G 1. Assin t olors rom 1 to P. Tn tr xists n L-olorin o G xtnin t olors on P, n totr 1 n orm n L-olorin o G, ontrition. Clim 5.5. I v V (P ) su tt V (P ) N[v], tn t surp o G inu y N(v) is not isomorpi to ny rp in {P 1, P, P, K }. 15

16 Proo. Suppos tt tr xists vrtx v V (P ) wr ll prolor vrtis r in N[v] n t surp G[N(v)] is isomorpi to surp in {P 1, P, P, K }. Sin N G [v] 4, tr xists n L-olorin o G[N[v]]. Sin (G, P, L, ) is miniml ountrxmpl, xtns to n L-olorin o G, wi in turn xtns to n L-olorin o G, ontrition. Clim 5.6. I v V (P ) s G (v), tn G (v) = n P is isomorpi to P 1, P, or P. Proo. By Clim 5., G (v) 1. I G (v) = n P = K, tn G[N G (v)] is isomorpi to P, ontritin Clim 5.5. Clim 5.7. P is isomorpi to P. 45 Proo. Suppos tt P is not isomorpi to itr P or K. I P is isomorpi to P 1, tn t 46 vrtx v o P s two istint niors u 1 n u tt r on t sm s v; lt U = {u 1, u }. 47 I P is isomorpi to P, tn som vrtx v in P s nior u 1 not in P tt srs 48 wit t in P ; lt U = {u 1 }. Lt P inu y V (P ) V (U). Noti P = n it 49 is isomorpi to P or K. Tr xists propr olorin o P tt xtns t olorin on P. But tn (G, P, L, ) s n(g) 1 4 n(p ) < n(g) 1 4n(P ), so tr xists n L-olorin o G tt 441 xtns, ontrition. 44 I P is isomorpi to K, w n rmov ny wit ot vrtis in P. By minimlity 44 o G, tr xists n L-olorin xtnin in G ut it is lso n L-olorin o G sin ot 444 npoints o v irnt olor in, ontrition Clim 5.8. I v V (G P ), tn G (v) 4. Proo. Suppos tt v V (G P ) s r (v). Tn G v is plnr rp wit no or 7-yl ontinin prolor surp P n list ssinmnt L. Sin (G, P, L, ) is minimum ountrxmpl, G v s n L-olorin. Howvr, v s t most tr niors n t lst our olors in t list L(v). Tus, tr is n xtnsion o t L-olorin o G v to n L-olorin o G, ontrition. Clim 5.4 lps us to prov t ollowin jnis o s. Clim 5.9. I 5-5 is jnt to trinl tn tr is -vrtx inint to ot o tm. Morovr, vry 5- is jnt to t most on trinl. Proo. Lt 5 5- oun y yl v 1, v, v, v 4, v 5. Lt - wit vrtis v 1 v x. Sin G s no or 6-yl, x {v, v 4, v 5 }. I x = v 4, tn Clim 5.4 implis v 1 v 4 v 5 n v v v 4 r lso trinulr s n w otin ontrition wit Clim 5.8 sin G is rp on 5 vrtis n only on 4 + -vrtx. By symmtry twn v n v 5 suppos tt x = v. Tn v is t sir -vrtx n w r on. Suppos tt 5 is jnt to two trinl s. E o tm s -vrtx in ommon wit 5. By symmtry ssum ts -vrtis r v n v 5. Tn v 1 v 4 n v v 4 r s n v 1 v v 4 is trinl jnt to 5 not srin ny -vrtx wit 5, wi is ontrition. Clim I two 4-s r jnt tn ty r ot inint to t sm -vrtx 16

17 Proo. Lt 1 n jnt 4-s oun y yls v 1, v, v, v 4 n v 1, v, x, x 1 rsptivly. Sin G os not ontin or 6-yls, 1 n must sr t lst vrtis. I ty sr our vrtis, w t ontrition wit Clim 5.8. By symmtry w ssum x 1 is v or v 4. I x 1 = v tn v, x, v n v 1, v, v 4 r trinulr s n w otin ontrition wit Clim 5.8. Hn x 1 = v 4 n v 1 s r two. Clim I 4- srs two or mor s wit trinulr s, tn it srs s wit xtly two. Morovr, tr is -vrtx v V (P ) inint to ot trinulr s n to. Proo. Lt 4- oun y yl v 1, v, v, v 4 n ssum tt v 1, v, x is trinulr. I x {v, v 4 } tn G woul violt Clim 5.4 or Clim 5.8. Hn x is not vrtx o t yl. Suppos or ontrition v, v 4, y is lso trinulr. Sin G os not ontin or 6-yls, x = y. By Clim 5.4, G s only 5 vrtis n ontrits Clim 5.8. Hn is jnt to t most two trinls. Assum tt v 4, v 1, y is trinulr. Sin G os not ontin or 6-yls, x = y. Tn v 1 is t sir -vrtx sin y Clim 5.8, v 1 V (P ). Clim 5.1. Evry -vrtx is jnt to t most two trinulr s. Proo. Lt v -vrtx jnt to tr trinulr s. Not tt ts r ll t s ontinin v. Tis ontrits tt P = P. 481 Sin G is miniml ountrxmpl, G os not ontin ny o t ruil oniurtions. 48 Spiilly, w us t t tt G vois (C) n (C4) (s Fiur ), wr no rmov vrtx 48 is prolor. 484 For v V (G) V (P ), p V (P ), n F (G) in initil r µ 0 (v) = (v) 4, 485 µ 0 (p) = (p) n µ 0 () = l() 4. By Eulr s Formul, t initil r sum is 8 + =. Sin δ(g P ) 4, t only lmnts o ntiv r r -s. Sin 487 or 6-yl is orin, δ(g P ) 4, n Clim 5.4, t lustrs (s Fiur 1) r trinls 488 (K), imons (K4), -ns (K5), 4-wls (K5), n 4-ns wit n vrtis intii (K5). 489 Spiilly not tt t 4-n (K6) ontins or 6-yl, so t most tr -s in 490 lustr sr ommon vrtx, unlss ty orm 4-wl (K5) n t ommon vrtx is t vrtx in t ntr o t wl. 49 Apply t ollowin isrin ruls, s sown in Fiur (R1) I p is -vrtx inint wit two 4-s, tn p sns r 9 to o tm. 494 (R) I is - n is n inint, tn lt t jnt to ross. (R) I is 5 + -, tn pulls r rom trou t. (R) I is 4- jnt to on -, tn lt 1,, n t otr s inint to For i {1,, }, lt i t jnt to ross i. For i {1,, }, t pulls r 1 9 rom t i trou t s n i. (R) I is 4- jnt to two -s, tn lt 1 n s o not inint to -s. For i {1, }, lt i t jnt to ross i. For i {1, }, t pulls r 1 18 rom t i trou t s n i. Lt v t vrtx sr y, n t otr -. Tn v sn r 9 to trou. 17

18 50 (R) Lt v 5 + -vrtx or prolor, n lt n inint -. (R) I v is 5-vrtx tt is not prolor, tn v sns r (R) I v is 6 + -vrtx or prolor, tn v sns r to. 9 to (R4) I X is lustr, tn vry - in X is ssin t vr r o ll -s in X. Noti tt prolor vrtis v similrly to 6 + -vrtis (R) (R) (R) (R) v 1 v p v (R1) (R) Fiur 9: Disrin ruls in t proo o Torm Noti tt t ruls prsrv t sum o t rs. Lt µ i (v) n µ i () not t r 509 on vrtx v or tr rul (Ri). W lim tt µ 4 (v) 0 or vry vrtx v n µ 4 () or vry ; sin t totl r sum is prsrv y t isrin ruls, tis ontrits 511 t ntiv r sum rom t initil r vlus. I v is 6 + -vrtx, tn y (R) v loss r 4 9 to inint -. Sin G vois or 6-yls, v is inint to t most 51 4 (v) -s. Tus µ 4(v) stisis µ 4 (v) (v) (v) (v) (v) = (v) Lt v 5 -vrtx not in P. I v is 4-vrtx, tn v is not involv in ny rul, so t rsultin r is 0. I v is 5-vrtx, tn y (R) v loss r 1 to inint -. 18

19 Sin G vois or 6-yls, v is inint to t most tr -s, so µ 4 (v) (v) 4 1 = (v) 5 = 0. Tror, µ 4 (v) 0 or vry vrtx v not in P. Lt v 5 -vrtx in P. I v is 5-vrtx or 4-vrtx tn rul (R) pplis t most (v) tims n µ 4 (v) (v) (v) > 0. 9 I v is -vrtx tn y Clim 5.1 (R) n (R) pply t most twi n µ 4 (v) (v) > 0. I v is -vrtx, tn t most on o (R1) n (R) pply n i (R) pplis, it pplis only on. Hn µ 4 (v) (v) = 0. Tror ll vrtis v V (G) v µ 4 (v) 0. Lt 4-. I (R) or (R) pplis to tn it must jnt to notr 4- n y Clim 5.10 n ty sr -vrtx v. Hn (R1) pplis to n v n t r lost in (R) n (R) is t most t r in in (R1). Tus, µ 4 () 0 or vry 4-. I is 6 + -, tn loss r t most 1 trou y (R), (R), or (R), so µ 4 () l() 4 1 l() = l() 4 0. Tror, µ 4 () 0 or vry Lt 5-. I is not jnt to -, loss no r y (R), ut oul los r usin (R) n (R), so µ 4 () l() l() = 8 l() I is jnt to -, y Clim 5.9 it is jnt to t most on n it srs t most two s wit it, so (R) is pplis t most twi wil t most r is lost trou o t 540 rminin tr s y (R) n (R) n w otin µ 4 () l() = 0. Tror, µ 4 () 0 i is 5-. All ojts tt strt wit nonntiv r v nonntiv r tr t isrin pross. It rmins to sow tt lustr o -s rivs nou r to rsult in nonntiv r sum. Osrv tt t ruls (R), (R), n (R) urnt tt i trinl is srin n wit 4 + -, tn rivs totl r 1 trou Cs 1: (K) Lt n isolt -. T tr jnt s 1,, n r ll 4 + -s. By (R), rivs r 1 trou inint, so µ 4() = = 0. 19

20 Cs : (K4) Lt 1 n -s in imon lustr (K4). Tn 1 is jnt to two s 1 n, n is jnt to two 4 + -s 1 n. By (R), t lustr rivs r trou o t our s on t ounry o t imon. Sin µ 0( 1 ) + µ 0 ( ) =, t r vlu on t imon tr rul (R) is. Sin G ontins no (C), tr is vrtx v inint to ot 1 n. I v is 5-vrtx, tn y (R), 1 n riv 554 r 1, n t rsultin r on t imon is zro. I v is 6+ -vrtx, tn y (R), n riv r 4 9, n t rsultin r on t imon is positiv. 556 Cs : (K5) Lt 1,, n -s in -n lustr (K5), wr is jnt to ot n. T initil r on tis lustr is. Tr r iv s on t ounry o tis lustr, so y (R) t lustr rivs r 5, rsultin in r 4 tr (R). Not tt 559 t is jnt to ot 1 n. Sin G ontins no (C), tr xists 5 + -vrtx v 560 inint to ot 1 n, n tr xists 5 + -vrtx u inint to ot n. I v u, tn y (R) v sns r t lst 1 to o 1 n n u sns r t lst 1 to 56 o n, rsultin in nonntiv r on t -n. I v = u n v is 6 + -vrtx, tn 56 y (R) v sns r 4 9 to 1,, n, rsultin in nonntiv r on t 564 -n. Otrwis, suppos tt v = u n v is 5-vrtx. Sin G ontins no (C4), tr xists notr 5 + -vrtx w inint to t lst on o 1 n. By (R) v sns r 1 to o 566 1,, n, n y (R) w sns r t lst 1 to t lst on o 1 n, rsultin in 567 nonntiv r on t -n. 568 Cs 4: (K5) Lt 1,,, n 4 -s in 4-wl (K5). T initil r on tis 569 lustr is 4. Tr r our s on t ounry o tis lustr, so y (R) t lustr rivs r 4, rsultin in r 8 tr (R). Lt v t 4-vrtx inint to ll our -s. 571 Lt u 1, u, u, n u 4 t vrtis jnt to v, orr ylilly su tt vu i u i+1 is t 57 ounry o t - i or i {1,, } n vu 4 u 1 is t ounry o 4. Sin G ontins no (C) n (v) = 4, u i is 5 + -vrtx. By (R), u i sns r t lst to t 574 lustr, rsultin in nonntiv totl r. 575 Cs 5: (K5) Lt 1,,, n 4 -s in 4-strip wit intii vrtis s in (K5). T 576 initil r on tis lustr is 4. Lt v, u 1, u, u, n u 4 t vrtis in t 4-strip, wr 577 v is inint to only 1 n 4, u 1 is inint to only 1 n, u is inint to,, n 4, 578 u is inint to 1,, n, n u 4 is inint to only n 4. Tr r six s on t ounry o tis lustr, so y (R) t lustr rivs r 6, rsultin in r 6 = 580 tr (R) Sin n orm imon, n G ontins no (C), on o u n u is 5 + -vrtx. Witout loss o nrlity, ssum u is 5 + -vrtx. Sin n 4 orm imon, n G ontins no (C), on o u n u 4 is 5 + -vrtx. I u is 5 + -vrtx, tn y (R), t lustr rivs r t lst + rom u n u, wi rsults in nonntiv totl r. Otrwis, u is 4-vrtx n u 4 is 5 + -vrtx. I u is 6 + -vrtx, tn y (R), t lustr rivs r t lst 4 + rom u n u 4. I u is 5-vrtx, tn sin 1 n orm imon n G ontins no (C4), on o v n u 1 is 5 + -vrtx. By (R), t lustr rivs r t lst + + rom u n u 4 n on o v n u 1. In itr s, t inl r is nonntiv. W v vrii tt t totl r tr isrin is nonntiv, ontritin t ntiv initil r sum. Tus, miniml ountrxmpl os not xist n vry plnr rp 0

21 wit no or 6-yl is (4, )-oosl. 6 No Cor 7-Cyl Torm 6.1. I G is pln rp not ontinin or 7-yl, tn G is (4, )-oosl. W prov t ollowin strntn sttmnt: Torm 6.. Lt G pln rp wit no or 7-yl, n lt P surp o G, wr P is isomorpi to on o P 1, P, P, or K, n ll vrtis in V (P ) r inint to ommon. Lt L (4, )-list ssinmnt o G P n lt propr olorin o P. Tr xists n xtnsion o to propr olorin o G su tt (v) L(v) or ll v V (G P ). 600 Proo. Suppos tt tr xists ountrxmpl. Slt ountrxmpl (G, P, L, ) y minimizin n(g) 1 4n(P ) mon ll or 7-yl r pln rps, G, wit surp P isomorpi 60 to rp in {P 1, P, P, K }, propr olorin o P, n (4, )-list ssinmnt L o G P 60 su tt os not xtn to n L-olorin o G. W will rr to t vrtis o P s prolor 604 vrtis. 605 Clim 6.. G is -onnt Proo. I G is isonnt, tn onnt omponnt n olor sprtly. Suppos tt G s ut-vrtx v. Tn tr xist onnt surps 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 n ssum witout loss o nrlity tt G 1 ontins t lst on vrtx o P, so lt S 1 t surp o P ontin in G 1. Lt S = {v} (V (G ) V (P )). Sin (G, P, L, ) is miniml ountrxmpl, tr is n L-olorin 1 o G 1 tt xtns t olorin on S 1. Usin t olor prsri y 1 on v, tr xists n L-olorin o G tt xtns t olorin on S. T olorins 1 n orm n L-olorin o G, ontrition. Clim 6.4. G s no sprtin -yls. Proo. Suppos tt P = v 1 v v is sprtin -yl o G. Lt G 1 t surp o G ivn y t xtrior o P lon wit P, n lt G t surp o G ivn y t intrior o P lon wit P. Sin P is sprtin, n(g 1 ) < n(g) n n(g ) < n(g). Sin t vrtis in P sr ommon, w n ssum witout loss o nrlity tt V (P ) V (G 1 ). Sin (G, P, L, ) is miniml ountrxmpl, tr xists n L-olorin 1 o G 1. Assin t olors rom 1 to P. Tn tr xists n L-olorin o G xtnin t olors on P, n totr 1 n orm n L-olorin o G, ontrition. Clim 6.5. I v V (P ) su tt V (P ) N[v], tn t surp o G inu y N(v) is not isomorpi to ny rp in {P 1, P, P, K }. Proo. Suppos tt tr xists vrtx v V (P ) wr ll prolor vrtis r in N[v] n t surp G[N(v)] is isomorpi to surp in {P 1, P, P, K }. Tn onsir t rp G = G v. Sin N G [v] 4, tr xists n L-olorin o G[N[v]]. Sin (G, P, L, ) is miniml ountrxmpl, xtns to n L-olorin o G, wi in turn xtns to n L-olorin o G, ontrition. 1

22 Clim 6.6. I v V (P ) s G (v), tn G (v) = n P is isomorpi to P 1, P, or P. Proo. By Clim 6., G (v) 1. I G (v) = n P = K, tn G[N G (v)] is isomorpi to P, ontritin Clim 6.5. Clim 6.7. P is isomorpi to on o P or K. 6 Proo. Suppos tt P is not isomorpi to itr P or K. I P is isomorpi to P 1, tn t 64 vrtx p o P s two niors u 1 n u tt r on t sm s p; lt U = {u 1, u }. I P is 65 isomorpi to P, tn som vrtx v in P s nior u 1 not in P tt srs wit t 66 in P ; lt U = {u 1 }. Lt P inu y V (P ) V (U). Noti P = n it is isomorpi 67 to P or K. Tr xists propr olorin o P tt xtns t olorin on P. But tn (G, P, L, ) s n(g) 1 4 n(p ) < n(g) 1 4n(P ), so tr xists n L-olorin o G tt xtns 69, ontrition Clim 6.8. I v V (G P ), tn G (v) 4. Proo. Suppos tt v V (G P ) s r (v). Tn G v is plnr rp wit no or 7-yl ontinin prolor surp P n list ssinmnt L. Sin (G, P, L, ) is minimum ountrxmpl, G v s n L-olorin. Howvr, v s t most tr niors n t lst our olors in t list L(v). Tus, tr is n xtnsion o t L-olorin o G v to n L-olorin o G, ontrition. Osrv tt n(g) 4. Rll tt in oniurtion (C, X, x), n L-olorin o V (C) \ X xtns to ll o C. Bus o tis t, i G ontins ruil oniurtion (C, X, x), tn tr is prolor vrtx in t st X, or ls G X s n L-olorin tt xtns to ll o G. Spiilly, w will us t t tt G vois (C), (C), (C4), (C5), (C6), (C7), n (C8). For v V (G) n F (G) in µ 0 (v) = (v) 4 + δ(v) n µ 0 () = l() 4 + ε(), wr δ(v) {0, 1} s vlu 1 i n only i v V (P ), n ε() {0, 1} s vlu 1 i n only i t ounry o is t st o prolor vrtis, V (P ). By Eulr s Formul, t initil r sum is t most 1. Clims 6.6 n 6.8 ssrt tt t only ntivly-r ojts r -s. For vrtx v, lt t k (v) not t numr o k-s inint to v. Apply t ollowin isrin ruls. Lt µ i (v) n µ i () not t r on vrtx v or tr rul (Ri). (R0) I v is prolor vrtx n is n inint - wit ntiv initil r, tn v sns r 1 to. (R1) I is - n is n inint, tn lt t jnt to ross. (R1) I is 5 + -, tn pulls r rom trou t. (R1) I is 4- n is t only - jnt to, tn lt 1,, n t otr s inint to. For i {1,, }, lt i t jnt to ross i. For i {1,, }, t pulls r 1 8 rom t i trou t s n i.

23 8 v 1 v 4 9 (R1) (R) (R) (R1) (R1), Cs 1 (R1), Cs Fiur 10: Disrin ruls (R1) n (R) in t proo o Torm (R1) I is 4- n is jnt to two -s 1 n (sy 1 = ), tn lt 1 n t otr s inint to, wr t s 1 n srin ts s r 6 + -s. For i {1, }, t pulls r 16 rom t i trou t s n i. (R) Lt v 5 + -vrtx wit v / V (P ) n lt n inint -. (R) I v is 5-vrtx, tn v sns r 1 to, wn = mx{, t (v)}. (R) I v is 6 + -vrtx, tn v sns r 1 to. (R) I is 6- wit µ () < 0 n v is n inint 5 + -vrtx or n inint vrtx in V (P ) wit µ 0 (v) > 0, tn v sns r 1 4 to. W lim tt µ (v) 0 or vry vrtx v n µ () 0 or vry. Sin t totl r sum ws prsrv urin t isrin ruls, tis ontrits t ntiv r sum rom t initil r vlus. Not tt 6-s r not inint to -s sin G os not ontin or 7-yl n sprtin -yls. Osrv tt 6- s µ 1 () < 0 i n only i ll s jnt to r 4-s, n o tos 4-s s two jnt -s. Clim 6.9. Lt v vrtx in V (P ). Tn µ (v) 0. In ition, i v is inint to 6- wit µ 1 () < 0, tn µ 0 (v) > 0. Proo. By Clims 6.6 n 6.7, w v µ 0 (v) = (v) 0. Not tt i µ 0 (v) 1 t (v) t 6(v), tn t inl r µ (v) is nonntiv. Sin (v) t (v) + t 6 (v), it suis to sow tt µ 0 (v) 1 4 (v) t (v).

24 Cs 1: P = P. Lt v 1, v, n v t vrtis in t -pt P. For i {1,, }, µ 0 (v i ) = (v i ). Sin P is not isomorpi to K, ts vrtis o not orm yl, n t to wi ll vrtis r inint is not -. Hn t (v i ) (v i ) 1. I (v i ) 4, tn µ 0 (v i ) = (v i ) 1 (v i) > 1 4 (v i) t (v i ). I (v ) =, tn µ 0 (v i ) = 0. Vrtx v is not inint to ny -s sin v 1 n v r not jnt. Morovr, v is not inint to ny 6- wit µ 1 () < 0. I su xist, v woul inint lso to 4- tt is inint to two trinls. Tis oniurtion o s rsults in sprtin trinl, or 7-yl or ontrition wit Clim I (v i ) = or i {1, }, tn µ 0 (v i ) = 0. I v i is jnt to -, tn lt v i t 694 nior o v i not in V (P ). Lt P t surp inu y (V (P ) {v i })\{v i}, wi orms 695 opy o P or K in G v i. For ny olor (v i ) L(v i ) \ {(v i)}, tr xists n L-olorin o 696 G v i s (G v i, P, L, ) is not ountrxmpl; tis olorin xtns to n L-olorin o G. 697 Tus, t (v i ) = 0. I v i is inint to 6- wit µ 1 () < 0, tn t otr inint to 698 v i is 4- tt is jnt to two -s. Tis rsults in or 7-yl, ontrition; 699 tus (R) os not pply to v i I (v i ) =, Clim 6.4 ssrts tt G s no sprtin -yls, so tn v i loss r t most 1 in (R0). I v i is inint to 6- wit µ 1 () < 0, tn t otr two s inint to v i r 4-s n ts 4-s r jnt to two -s. Tis rts or 7-yl, ontrition, so (R) os not pply to v i n µ (v i ) Cs : P = K. Lt v 1, v, n v t vrtis in t -yl P, so µ 0 (v i ) = (v i ) or v i. By Clim 6.4, G s no sprtin -yl, so t tr vrtis r inint to ommon - wit µ 0 () = 0. Tror, vrtx v i sns r 1 to t most (v i) 1 inint -s y (R0). Rll tt (v i ) y Clim 6.6. Suppos tt (v i ) =. I t (v i ) > 1, t surp o G inu y t nioroo o v i is isomorpi to P or K, ontritin Clim 6.5. I (v i ) 4, tn µ 0 (v i ) = (v i ) 1 (v i) 1 4 (v i) t (v i ). Tror, µ (v i ) 0. Tus, in ll ss prolor vrtx v s µ (v) W will now sow tt ll ojts tt strt wit nonntiv r lso n wit nonntiv 71 r. 71 I is 4-, tn (R1) n (R1) o not pull r rom, sin tis woul rquir 714 to jnt to 4- tt is jnt to - t, ut tn,, n t ontin or yl. Tus, µ () = 0 or vry I is 5-, tn sin G ontins no or 7-yls, is not jnt to two -s n is not jnt to 4-. Tror, loss r t most 8 y (R1), ut loss no r 718 usin (R1), so µ () > 0 or vry I is 6-, tn is not jnt to - sin G ontins no or 7-yl. Osrv 70 tt y Clim 6. t ounry o is simpl 6-yl. So i sns r trou n urin (R1), it n sn r 1 8 trou y (R1), or it n sn r 8 trou y (R1). T only wy tt tis will rsult in ntiv r tr (R1) n (R) is or to sn r 7 trou o its six s y (R1); tis will us µ () = 6 8 = 1 4. I s prolor 74 vrtx v on its ounry, tn y Clim 6.9, v s positiv r tr (R0); y (R), rivs 75 r t lst 1 4, rsultin in µ () 0. I s no inint prolor vrtis, tn sin G 76 ontins no (C), som vrtx v on t ounry o is 5 + -vrtx. By (R) v sns r 1 4 to 8 4

25 n n µ () 0. Osrv t ollowin lim onrnin t strutur out vrtx tt loss r y (R). Clim Lt v 5 + -vrtx wit t tr inint s 1,, n, in yli orr. I v sns r to y (R), tn 1 n r 4-s n is 6-. I is 7 + -, tn y (R1) loss r t most 8 µ () l() 4 8 l() = 5 l() 4 > 0. 8 trou. Tus, 7 Tror, µ () > 0 or vry Nxt, w will onsir vrtx v not in V (P ). 75 I v is 4-vrtx, tn v os not los r y ny rul, so t rsultin r is I v is 5-vrtx, lt = mx{, t (v)} n v loss r 1 t (v) to inint -s y (R). 77 I (R) os not pply to v, tn v sns r t most 1 to inint -s n µ (v) 0. I 78 (R) pplis to v, tn v is inint to s 1,, n wr 1 n r 4-s n is Sin (v) = 5 n G s no or 7-yl, t rul (R) pplis t most on. In, i 740 (R) woul pply twi, tn v woul inint to two 4-s srin n n o ts 741 two 4-s srs two s wit trinls n tis ivs or 7-yl. I (R) pplis on, 74 tn t (v) n v loss r t most y (R) n r 1 4 y (R), so µ (v) I v is 6 + -vrtx, tn lt k = t (v) n l t numr o tims (R) pplis to v. Noti tt k 4 5(v) sin G vois or 7-yls. Furtr, noti tt k + l (v), sin tt ins r rom v y (R) is pr y 4- in t yli orr o s roun v. By (R), v n los r 1 to inint -, n v n los r t most 1 4 to 747 inint 6- y (R). Tn v ns wit r µ (v) (v) 4 1 k 1 4 l. I (v) = 6, tn osrv k + l 4 n n µ (v) 0. I (v) = 7, tn, k, n l stisy t ollowin linr prorm wit ul on vrils 1,, n : 751 min 1 k 1 4 l s.t k 0 k l 0, k, l 0 mx 7 1 s.t ,, 0 T ul-sil solution ( 1,, ) = ( 40, 1 0, 1 ) 4 monstrts tt 1 k 1 4 l 7 40 > 4, n 75 tus µ (v) > 0 or vry 7 + -vrtx. 754 It rmins to sown tt t lustrs riv nou r to om nonntiv. Sin G 755 ontins no sprtin -yl, G os not ontin t lustr (K5) or t lustrs (K6) (K6r). 756 Osrv tt tr is no prolor vrtx v o r t most tr wr ll s inint to 757 v v lnt tr. Finlly, it is wort notin in tt i G ontins ruil oniurtion 758 (C, X, x), tn tr is prolor vrtx in t st X. 759 I vrtx v is 5 + -vrtx or v V (P ), w sy v is ull; i v is 6 + -vrtx or v V (P ), tn v is vy. Not tt vy vrtx v sns r 1 to inint ntivly-r - y (R0) or (R). I P 761 = K, w ll P t prolor. 5

26 v 1 1 (K) (K4) (K5) Fiur 11: Clustrs (K), (K4), n (K5) 76 Cs 1: (K) Lt t isolt - in (K). I is t prolor, tn µ () = µ 0 () = 0. Otrwis, t initil r on is 1. By (R1), rivs r trou its ounry 764 s, rsultin in nonntiv inl r. 765 Cs : (K4) Lt 1 n -s in imon lustr (K4). First, suppos witout loss o 766 nrlity tt 1 is t prolor. T initil r o t lustr is 1. Tn rivs r 1 y (R0) n r 8 y (R1), rsultin in positiv inl r. Otrwis, t 8 trou o t two s on t ounry o t lustr, rsultin in r 1. I t lustr ontins prolor vrtx u, tn it rivs r y (R0). Otrwis, sin G ontins no (C), tr is 5+ -vrtx v inint to ot 1 n. By (R), tis vrtx sns r t lst 1 to o t 77 s, rsultin in nonntiv inl r. 77 Cs : (K5) Lt 1,, n -s in -n lustr (K5), wr is jnt to ot n. Suppos tt t lustr ontins prolor, so t initil r on t lustr 775 is. I is prolor, tn t lustr rivs r 4 1 y (R0); i 1 or is prolor, tn t lustr rivs r 1 y (R0) n r 8 y (R1). In itr s, t inl 777 r is nonntiv. 778 I P = K or t lustr os not ontin t prolor, tn t initil r on t lustr is. By (R1), t lustr rivs r 5 8, rsultin in r 9 8. Not tt t 780 s 1 n orm imon n t s n orm imon. Sin G ontins no 781 (C), tr xists ull vrtx v inint to ot 1 n. Similrly, tr xists ull vrtx 78 u inint to n. I u v, tn y (R0) or (R), v sns r t lst 1 to o 1 78 n n u sns r t lst 1 to o n, rsultin in nonntiv r on 784 t lustr. I u = v n v is vy vrtx, tn v sns r 1 to 1,, n, 785 rsultin in nonntiv r on t lustr. Otrwis, suppos tt u = v / V (P ) n v is vrtx. Sin G ontins no (C4), tr xists notr ull vrtx w tt is inint to t 787 lst on o 1 n. By (R), v sns r 1 to 1,, n, n y (R0) or (R), w 788 sns r t lst 1 to on o 1 n, rsultin in nonntiv r on t lustr. u 1 u 1 v u 4 v u 5 u 4 u u 1 u 1 (K5) (K6) (K6) Fiur 1: Clustrs (K5), (K6), n (K6) u 4 6

27 789 Cs 4: (K5) Lt 1,,, n 4 -s in 4-wl (K5). I t lustr ontins prolor, tn t initil r on t lustr is ; t lustr rivs r 5 1 y (R0) n r 8 y (R1), rsultin in positiv inl r. Otrwis, t initil r 8, rsultin in r 5. Lt v 79 t 4-vrtx inint to ll our -s. Lt u 1, u, u, n u 4 t vrtis jnt to v, 794 orr ylilly su tt vu i u i+1 is t ounry o t - i or i {1,, } n vu 4 u 1 is 795 t ounry o 4. Sin t lustr os not ontin t prolor, v is not prolor 796 vrtx. Sin G ontins no (C), u i is ull vrtx. Wn u i is 5-vrtx, it is inint to 797 two 7 + -s, so u i sns r 1 to inint - y (R). Tus, u i sns r 798 t lst 1 to t lustr y (R0) or (R), rsultin in nonntiv inl r. 799 Cs 5: (K6) Lt 1,,, n 4 -s in 4-strip lustr (K6). I t lustr ontins 800 t prolor, tn t initil r on t lustr is. I 1 or 4 is prolor, tn t lustr rivs r y (R0) n r 4 8 y (R1); i or is prolor, tn t lustr rivs r y (R0) n r 5 8 y (R1). In itr s, t rsultin 80 inl r is nonntiv. I t lustr os not ontin t prolor, tn t initil r on tis lustr is 4. By (R1), t lustr rivs r 6 8, rsultin in r Not tt or i {1,, }, t s i n i+1 orm imon. Sin G ontins no (C), 806 tr xists ull vrtx v inint to ot i n i+1. Lt u 1 ull vrtx inint to 807 n. Witout loss o nrlity, u 1 is not inint to 4, so tr is ull vrtx u inint to 1 n. I u 1 is vy vrtx, t lustr rivs r rom u 1 y (R0) or (R), n r t lst rom u y (R0) or (R), rsultin in positiv inl r. Otrwis, u 1 is 5-vrtx, so u 1 sns r 1 y (R), rsultin in r I u is inint to, tn u sns r t lst y (R0) or (R), rsultin in positiv inl r. Otrwis, u is inint wit 1 n ut not. I u is lr vrtx, it sns r y (R0) or (R). Otrwis, sin G ontins nitr (C) or (C4), tr is tir ull vrtx u. T lustr rivs r 1 rom u y (R) n r t lst 1 rom u y (R0) or (R). In s, t rsultin inl r is nonntiv. 816 Cs 6: (K6) Lt 1,,, n 4 -s in 4-n lustr (K6). Lt v t ntr o 817 t n, wit niors u 1, u, u, u 4, n u 5 wr or i {1,, }, i n i+1 r jnt on 818 t vu i+1. I t lustr ontins t prolor, tn t initil r on t lustr is. I 1 or 4 is prolor, tn t lustr rivs r 4 1 y (R0) n r 4 8 y (R1); i or is prolor, tn t lustr rivs r 5 1 y (R0) n r 5 8 y 81 (R1). In itr s, t rsultin inl r is positiv. 8 I t lustr os not ontin t prolor, tn t initil r on tis lustr is 4. By (R1), t lustr rivs r 6 8, rsultin in r I v is vy vrtx, tn y (R0) or (R) v sns r to t lustr, rsultin in positiv r. Otrwis, 85 v / V (P ) n v is 5-vrtx, so v sns r 1 to t lustr y (R), rsultin in r 4. I tr is vy vrtx in {u, u, u 4 }, tn tt vrtx ontriuts r 1 86 to t 87 lustr, rsultin in positiv r. I tr is no vy vrtx in {u, u, u 4 }, tn tr is 88 t lst on 5-vrtx in {u, u, u 4 } sin G ontins no (C4). I tr r multipl 5-vrtis in {u, u, u 4 }, tn sns r 1 89 to t lustr y (R), rsultin in positiv r. I 80 tr is only 5-vrtx w mon u, u, n u 4, tn tr is ull vrtx z {u 1, u 5 } sin G os not ontin (C4) or (C5); t lustr rivs r 1 rom w y (R) n t lst rom z y (R0) or (R), rsultin in positiv inl r. 7