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

Transcription

1 (4, )-ooslty o plnr rps wt orn struturs Znr Brkkyzy 1 Crstopr Cox Ml Dryko 1 Krstn Honson 1 Mot Kumt 1 Brnr Lký 1, Ky Mssrsmt 1 Kvn Moss 1 Ktln Nowk 1 Kvn F. Plmowsk 1 Drrk Stol 1,4 Dmr 11, 015 Astrt All plnr rps r 4-olorl n 5-oosl, wl som plnr rps r not 4- oosl. Dtrmnn w proprts urnt tt plnr rp n olor usn lsts o sz our s rv snnt ttnton. In trms o onstrnn t strutur o t rp, or ny l {, 4, 5, 6, 7}, plnr rp s 4-oosl t s l-yl-r. In trms o onstrnn t lst ssnmnt, on rnmnt o k-ooslty s ooslty wt sprton. A rp s (k, s)-oosl t rp s olorl rom lsts o sz k wr jnt vrts v t most s ommon olors n tr lsts. Evry plnr rp s (4, 1)-oosl, ut tr xst plnr rps tt r not (4, )-oosl. It s n opn quston wtr plnr rps r lwys (4, )-oosl. A or l-yl s n l-yl wt on tonl. W monstrt or l {5, 6, 7} tt plnr rp s (4, )-oosl t os not ontn or l-yls. 1 Introuton A propr olorn s n ssnmnt o olors to t vrts o rp G su tt jnt vrts r ssn stnt olors. A (k, s)-lst ssnmnt L s unton tt ssns lst L(v) o k olors to vrtx v so tt L(v) L(u) s wnvr uv E(G). A propr olorn φ o G su tt φ(v) L(v) or ll v V (G) s ll n L-olorn. W sy tt rp G s (k, s)-oosl, or ny (k, s)-lst ssnmnt L, tr xsts n L-olorn o G. W ll ts vrton o rp olorn ooslty wt sprton. Not tt wn rp s (k, k)-oosl, w smply sy t s k-oosl. Osrv tt G s (k, t)-oosl, tn G s (k, s)-oosl or ll s t. A notl rsult rom Tomssn [11] stts tt vry plnr rp s 5-oosl, so t ollows tt ll plnr rps r (5, s)-oosl or ll s 5. Forn rtn struturs wtn plnr rp s ommon rstrton us n rp olorn. Torm 1. summrzs t urrnt knowl on (, 1)-ooslty o plnr rps. Škrkovsk [1] onjtur tt ll plnr rps r (, 1)-oosl; ts quston s stll opn n s prsnt low s Conjtur Dprtmnt o Mtmts, Iow Stt Unvrsty, Ams, IA, U.S.A. {znr,mryko,kons,mkumt,lky,kymss,kmoss,knowk,kplmow,stol}@stt.u Dprtmnt o Mtmtl Sns, Crn Mllon Unvrsty, Pttsur, PA, U.S.A. oox@nrw.mu.u Support y NSF rnt DMS Dprtmnt o Computr Sn, Iow Stt Unvrsty, Ams, IA, U.S.A. 1

2 Conjtur 1.1 (Škrkovsk [1]). I G s plnr rp, tn G s (, 1)-oosl. Torm 1.. A plnr rp G s (, 1)-oosl G vos ny o t ollown struturs: - -yls (Krtovíl, Tuz, Vot [9]). - 4-yls (Co, Lký, Stol [4]). - 5-yls n 6-yls (Co, Lký, Stol [4]). In ts ppr, w ous on 4-ooslty wt sprton. Krtovíl, Tuz, n Vot [9] prov tt ll plnr rps r (4, 1)-oosl, wl Vot [1] monstrt tt tr xst plnr rps tt r not (4, )-oosl. It s not known ll plnr rps r (4, )-oosl. Conjtur 1. (Krtovíl, t l. [9]). I G s plnr rp, tn G s (4, )-oosl. Torm 1.4 (Krtovíl, t l. [9]). I G s plnr rp, tn G s (4, 1)-oosl. Torm 1.4 ws strntn y Krst n Lký [8], wr t s sown tt w n llow n npnnt st o vrts to v lsts o sz rtr tn 4. Torm 1.5 (Krst n Lký [8]). Lt G plnr rp n I V (G) n npnnt st. I L ssns lsts 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-olorn. In ton to t work summrz ov, tr r svrl rsults rrn 4-ooslty. A rp s k-nrt o ts surps s vrtx o r t most k. Eulr s ormul mpls plnr rp wt no -yls s -nrt n n 4-oosl. Ts n otr smlr rsults r lst low n Torm 1.6. For t lst rsult n Torm 1.6, not tt or l-yl s n l-yl wt n tonl onntn two o ts non-onsutv vrts. Torm 1.6. A plnr rp G s 4-oosl G vos ny o t ollown struturs: - -yls (olklor). - 4-yls (Lm, Xu, Lu, [10]). - 5-yls (Wn n L [14]). - 6-yls (Fjvz, Juvn, Mor, n Škrkovsk [7]). - 7-yls (Frz [6]). - Cor 4-yls n or 5-yls (Boron n Ivnov []). Our mn rsults n ts ppr r lst low n Torm 1.7. Not tt ouly-or l-yl s or l-yl wt n tonl. Torm 1.7. A plnr rp G s (4, )-oosl G vos ny o t ollown struturs: - Cor 5-yls. - Cor 6-yls. - Cor 7-yls. - Douly-or 6-yls n ouly-or 7-yls. W prov s o Torm 1.7 sprtly. In Ston 4, w or or 5-yls (s Torm 4.1). In Ston 5, w or or 6-yls (s Torm 5.1); w us prts o ts proo to lso prov t s wn orn ouly-or 6-yls n ouly-or 7-yls (s Corollry 5.). In Ston 6, w or or 7-yls (s Torm 6.). Tr r mny turs ommon to ll o ts proos, w w tl n Stons n.

3 1.1 Prlmnrs n Notton Rr to [15] or stnr rp tory trmnoloy n notton. Lt G 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 omplt rp, yl rp, n pt rp, rsptvly, on n vrts. T opn noroo o vrtx, not N(v), s t st o vrts jnt to v n G; t los noroo, not N[v], s t st N(v) {v}. T r o vrtx v, not G (v), s t numr o vrts jnt to v n G; w wrt (v) wn t rp G s lr rom t ontxt. I t r o vrtx v s k, w ll v k-vrtx; t r o v s t lst k, w ll v k + -vrtx. T lnt o, not l(), s t lnt o t ounry wlk. I t lnt o s k, w ll k-; t lnt o s t lst k, w ll k + -. Ovrvw o Mto All o our mn rsults us t srn mto. W rr t rr to t survys y Boron [] n Crnston n Wst [5] or n ntrouton to srn, w s mto ommonly us to otn rsults on plnr rps. For rl numrs v,,, w n ntl r vlus µ(v) = v (v) or vry vrtx v n ν() = l() or vry. I v > 0, > 0 n v + = > 0, tn Eulr s ormul mpls tt v µ(v) + ν() =, n t totl r on t ntr rp s ntv. W tn n srn ruls tt sr mto or movn r vlu mon vrts n s wl onsrvn t totl r vlu. W monstrt tt G s mnml ountrxmpl to our torm, tn vry vrtx n ns wt nonntv r tr t srn pross, w s ontrton. Intutvly, ts pross works wll wn orn strutur (su s sort or yl) wt low r. In Ston, w onrtly n rul onurtons. Loosly, rul onurton s strutur C n rp G wt (4, )-lst ssnmnt L wr ny L-olorn o G C xtns to n L-olorn o G. I w r lookn or mnml xmpl o rp tt s not (4, )-oosl, tn non o ts rul onurtons ppr n t rp. W n lr lst o onurtons, (C1) (C1) (s Fur ), n prov ty r rul usn vrous nr onstrutons. T onurtons (C1) (C10) r us wn orn or 6- or 7-yls, wl t onurtons (C9) (C1) r us wn orn or 5-yls. T us o rnt onurtons s u to rns n our srn rumnts. In Ston 4, w or or 5-yls n vry - s jnt to t most on otr -. Morovr, -s r not jnt to 4-s. Tus, our ntl r unton n ts s urnts tt t only ojts wt ntv ntl r r 4- n 5-vrts. In Stons 5 n 6, w us rnt srn strty. Our ntl r vlus urnt tt t only ojts o ntv r r -s. Tus, our srn ruls r sn to sn r rom 5 + -s n 4 + -vrts to -s. Howvr, s w or or 6-yls or or 7-yls, tr my mny -s vry los to otr. I G s pln rp n G s ts ul, tn lt F t st o -s o G n lt G t nu surp o G wt vrtx st F. A lustr s mxml st o -s tt r onnt n G,.., onnt omponnt o G. Not tt two -s srn n r jnt n G, n two -s srn only vrtx r not jnt n G. S Fur 1 or lst o t lustrs wt mxmum yl lnt sx n vry ntrnl vrtx o r t lst our. In ts urs, t outr yl s not nssrly l yl, ny r ll wt ry s not, n pr o squr vrts rprsnt snl vrtx. Atonlly, ol s sr sprtn

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) (K6) (K6j) (K6k) (K6l) (K6m) (K6n) (K6o) (K6p) (K6q) (K6r) Ts r ll o t possl lustrs wt lonst yl t most sx n mnmum r our. Bol s monstrt sprtn -yls. Gry rons snt yls tt r not s. W roup our lustrs y t lnt o t lonst yl n t lustr. Tus onurton (Kn) s mxmum yl lnt o n. Fur 1: Clustrs wt mxmum yl lnt t most sx. 4

5 -yls, w r yls n pln rp wos xtror n ntror rons ot ontn vrts not on t yl. Ts urs r s on t lst o lustrs us y Frz [6] n t proo tt 7-yl-r plnr rps r 4-oosl. For k {1, }, tr s xtly on wy to rrn k -s n lustr. A trnl s lustr ontnn xtly on -; s (K). A mon s lustr ontnn xtly two -s; s (K4). For k, tr r multpl wys to rrn k - n lustr. A k-n s lustr o k -s ll nnt to ommon vrtx o r t lst k + 1; s (K5) n (K6). A k-wl s lustr o k -s ll nnt to ommon vrtx o r xtly k; s (K5) n (K6). Not tt t vrtx nnt to ll s o -wl s r. A k-strp s lustr o k -s 1,..., k wr t ounrs o t -s r sjont xpt 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 -s n lustr, tn w wll prov tt t totl r on 1,..., k tr srn s nonntv. Tus, som o t -s my v ntv r, ut ts s ln y otr -s n t lustr vn postv r. Hn, our proos n wt lst o ll possl lustr typs n vryn tt s nonntv totl r. Wl tr r totl lustrs tt vo or 7-yls, w o not v tt mny ss to k. T lustrs (K5) n (K6) (K6r) v tr ol s, monstrtn sprtn - yl. W vo kn ts ss y usn strntn olorn sttmnt (s Torm 6.) tt llows our mnml ountrxmpl to not ontn ny sprtn -yls. Rul Conurtons In ts ston, w sr struturs tt nnot ppr n mnml ountrxmpl to Torm 1.7. Lt G rp, : V (G) N, n s nonntv ntr. A rp s -oosl G s L-oosl or vry lst ssnmnt L wr L(v) (v). An (, s)-lst-ssnmnt s lst ssnmnt 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) = uv E(G) n (u) = (v) = 1. A rp G s (, s)-oosl G s L-olorl or vry (, s)-lst-ssnmnt L. Dnton.1. A onurton s trpl (C, X, x) wr C s pln rp, X V (C), n x : V (C) {0, 1,, } s n xtrnl r unton. A rp G ontns t onurton (C, X, x) C pprs s n nu surp C o G, n or vrtx v V (C), tr r t most x(v) s n G rom t opy o v to vrts not n C. For trpl (C, X, x), n t lst-sz unton : V (C) N s { 4 x(v) v X (v) = 1 v / X. A onurton (C, X, x) s rul C s (, )-oosl. Not tt rp G wt (4, )-lst ssnmnt L ontns opy o rul onurton (C, X, x) n G X s L-oosl, tn G s L-oosl. Frst, w not tt (C, X, x) s rul onurton, tn ny wy to n twn stnt vrts o X n lowr tr xtrnl r y on rsults n notr rul onurton. 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 onurtons, s wt only on npont r xtrnl s. Vrts n X r ll wt wt. Fur : Rul onurtons. 6

7 (C1) (C) (C4) (C5) (C10) (C11) (C1) (C1) (C14) (C15) (C16) Fur : Alon-Trs Ornttons. Lmm.. Lt (C, X, x) rul onurton, n suppos tt x, y X r nonjnt vrts wt { x(x), x(y) 1. Lt (C, X, x ) t onurton wr C = C + xy, X = X, n x x(v) v / {x, y} (v) = x(v) 1 v {x, y},. Tn t onurton (C, X, x ) s rul. Proo. Lt t lst-sz unton or C n not tt C s (, )-oosl. Smlrly lt t lst-sz unton on t onurton (C, X, x ), n lt L 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 o L (x) n L (y) to S untl S =. Now lt S = {, } su tt L (x) n L (y), n n lst ssnmnt L on C y rmovn rom L (x) n rmovn rom L (y). Osrv tt L s n (, )-lst ssnmnt n n tr xsts n L-olorn o C. Sn L(x) L(y) =, ts propr L-olorn o C s lso n L -olorn o C. W wll us Lmm. mpltly y ssumn tt C[X] pprs s n nu surp n our mnml ountrxmpl G..1 Rulty Proos In ts ston, w prov tt onurtons (C1) (C1) sown n Fur r rul..1.1 Alon-Trs Torm W wll us t lrt Alon-Trs Torm [1] to qukly prov tt mny o our onurtons r rul. In t, onurtons tt r monstrt n ts wy r rul or 4-ooslty, not just (4, )-ooslty. A rp D s n orntton o rp G G s t unrlyn unrt rp o D n D s no -yls; lt + D (v) n D (v) t out- n n-r o vrtx v n D. An Eulrn surp o rp D s sust S E(D) su tt, or vry vrtx v V (D), t numr o outon s o v n S s qul to t numr o nomn s o v n S. Lt EE(D) t numr o Eulrn surps o vn sz n EO(D) t numr o Eulrn surps o o sz. 7

8 Torm. (Alon-Trs Torm [1]). Lt G rp n : V (G) N unton. Suppos tt tr xsts n orntton D o G su tt + D (v) (v) 1 or vry vrtx v V (G) n EE(D) EO(D). Tn G s -oosl. W ll n orntton n Alon-Trs orntton t stss t ypotss o Torm.. For onurton (C, X, x) n t ssot lst-sz unton, t sus to monstrt n Alon-Trs orntton o C wt rspt to. S Fur or lst o Alon-Trs ornttons o svrl onurtons. Corollry.4. T ollown onurtons v Alon-Trs ornttons n n r rul: (C1), (C), (C4), (C5), (C10), (C11), (C1), (C1), (C14), (C15), (C16)..1. Drt Proos In t proos low, w onsr onurton (C, X, x) wt lst-sz unton n ssum tt n (, )-lst-ssnmnt L s vn or C. W wll monstrt tt C s L-olorl. Rr to Fur or rwns o t onurtons. Frst rll t ollown t out lst-olorn o yls. Ft.5. I L s -lst ssnmnt o n o yl, tn tr os not xst n L-olorn o t yl n only ll o t lsts r ntl. Lmm.6. (C) s rul onurton. Proo. Lt v 1,..., v 4 t vrts o 4-yl wt or v v 4 n lt v n v 4 v xtrnl r 1; t olors (v 1 ) n (v ) r x. E o v n v 4 v t lst on olor n tr lsts otr tn (v 1 ) n (v ). Sn L(v ) or {, 4}, tr on o ts vrts s t lst two olors vll, or L(v ) L(v 4 ) = {(v 1 ), (v )}. In tr s, w n xtn t olorn. For t onurtons (C6), (C7), n (C8), ll t vrts s n Fur 4: ll t ntr vrtx v 0 n t outr vrts v 1,..., v 5, strtn wt t vrtx rtly ov v 0, movn lokws. 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 lls or onurtons (C6), (C7), n (C8). Lmm.7. (C6) s rul onurton. 8

9 Proo. T olors (v 1 ) n (v 4 ) r trmn. I (v 1 ) n (v 4 ) r ot n L(v 0 ), tn slt (v 5 ) rom L(v 5 )\(L(v 0 ) {(v 1 ), (v 4 )}); otrws, slt (v 5 ) L(v 5 )\{(v 1 ), (v 4 )} rtrrly. Dn L (v 0 ) = L(v 0 ) \ {(v 1 ), (v 4 ), (v 5 )}, L (v ) = L(v ) \ {(v 1 )}, n L (v ) = L(v ) \ {(v 4 )} n not tt L (v ) or ll {0,, }. I L (v 0 ) = L (v ) =, tn L (v 0 ) L (v ), so t -yl v 0 v v s n L -olorn y Ft.5. Lmm.8. (C7) s rul onurton. Proo. I tr xsts olor L(v 1 ) L(v 4 ), strt y ssnn (v 1 ) = (v 4 ) = ; tn rly olor t rmnn vrts 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 ) =. Slt olor (v 4 ) L(v 4 ). Consrn v 4 s n xtrnl vrtx n norn t v 1 v 5, t 4-yl v 0 v 1 v v orms opy o (C4), w s rul y Corollry.4. Tus, tr xsts n L-olorn o v 0,..., v 4 ; ts olorn xtns to v 5 sn L(v 1 ) L(v 5 ) =. I L(v 4 ) L(v 5 ) =, tn tr xsts n L-olorn y symmtr rumnt. Otrws, tr xst olors L(v 1 ) \ L(v 5 ) n L(v 4 ) \ L(v 5 ); ssn (v 1 ) = n (v 4 ) =. Slt (v ) L(v ) \ {(v 1 )}. Dn L (v 0 ) = L(v 0 ) \ {(v 1 ), (v ), (v 4 )} n L (v ) = L(v ) \ {(v ), (v 4 )}. Not tt L (v 0 ) = L (v ) = 1, tn L(v 0 ) L(v ) = {(v ), (v 4 )} n n L (v 0 ) L (v ) =. Tus, t olorn xtns y rly olorn v, v 0, n v 5. Lmm.9. (C8) s rul onurton. Proo. I L(v 1 ) L(v ) =, tn rly olor v n v ; wt rmns s (C4) n t olorn 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. Slt (v 1 ) L(v 1 ) \ L(v ), (v ) L(v ) \ L(v ). Dn L (v 0 ) = L(v 0 ) \ {(v 1 ), (v )}, L (v 4 ) = L(v 4 ) \ {(v )}, n L (v 5 ) = L(v 5 ) \ {(v )}. Osrv tt w n L -olor t -yl v 0 v 4 v 5 y Ft.5 n tn slt (v ) L(v ) \ {(v 0 )}. I tr xsts olor L(v 1 ) L(v ), strt y ssnn (v 1 ) = (v ) = n tn ssn (v ) L(v )\{}. Dn L (v 0 ) = L(v 0 )\{, (v )}, L (v 4 ) = L(v 4 )\{}, n L (v 5 ) = L(v 5 )\{}. Osrv tt t -yl v 0 v 4 v 5 s n L -olorn y Ft.5. Lmm.10. (C9) s rul onurton. Proo. Consr t vrtx v o rtrry xtrnl r n lt (v) t olor ssn to v. Lt u 1 n u t two nors o v n t onurton. I w rmov (v) rom t lsts on u 1 n u, osrv tt t lst two olors rmn n vry lst or vry vrtx o t 5-yl. I tr s no L-olorn o t onurton, tn Ft.5 ssrts tt ll lsts v sz two n ontn t sm olors; owvr, ts mpls tt L(u 1 ) = L(u ) n L(u 1 ) L(u ) =, ontrton..1. Tmplt Conurtons T onurtons (C17) (C1) r spl ss o nrl onstrutons ll tmplt onstrutons. Lt (C, X, x) onurton wt vrts u, v X. A uv-pt P s ll spl uv-pt ll ntrnl vrts o P v r two n C n xtrnl r two. A uv-pt P s ll n xtr-spl uv-pt ll ntrnl vrts o P v xtrnl r two n r two n C, xpt or onsutv pr xy wr x(x) = x(y) = 1, (x) = (y) =, n tr s vrtx 9

10 z / X su tt z s ommon nor to x n y, n z s not jnt to ny otr vrts n C. Usn ts spl n xtr-spl pts, w n sr svrl onurtons y t ollown tmplts (s Fur 5), onsstn o (B1) trnl uvw, wr x(u) = x(w) =, x(v) = 0, n xtr-spl uv-pt P 1, n spl vw-pt P, n (B) trnl vwr, wr x(r) =, x(w) = 1, x(v) = 0, vrtx u jnt to v wr x(u) =, n xtr-spl uv-pt P 1, n spl vw-pt P. z P 1 x y u v (B1) w P P 1 x u z r y v (B) Dott lns nt spl pts or xtr-spl pts. Vrts n X r ll wt wt. Fur 5: Tmplts or rul onurtons. W mk som s osrvtons out spl n xtr-spl pts tt wll us to prov tt ts tmplts orrspon to rul onurtons. Lt P spl uv-pt or n xtr-spl uv-pt. For vry olor L(u), lt P u () t st ontnn olor L(v) su tt ssnn (u) = n (v) = os not xtn to n L-olorn o P. Sn w n rly olor P strtn t u untl rn v, tr s t most on olor n P u (). Furtr, u P () n only ts ry olorn pross s xtly on o or vrtx n P. Tus, P u () = {} tn lso v P () = {}. Sn L s n (, )-lst ssnmnt, jnt vrts v t most two olors n ommon. Tus, tr r t most two olors 1, L(u) su tt P u ( ). Morovr, osrv tt tr r two stnt olors 1, L(u) su tt P u ( ), tn ot 1 n r n vry lst lon P n n { 1, } L(v). I P s n xtr-spl uv-pt wt -yl xyz wr xy s n t pt P, tn tr olor s ssn to z (s x(z) = ) tr on o x or y s tr olors vll or L(x) L(y) 1. Tror, P s n xtr-spl uv-pt, tn tr s t most on olor L(u) su tt P u (). Lmm.11. All onurtons mtn t tmplt (B1) r rul. Proo. Lt (C, X, x) onurton mtn t tmplt (B1) n lt L n (, )-lst ssnmnt. Lt L(u) = { 1, }. Sn P 1 s n xtr-spl pt, tr s t lst on {1, } su tt u P 1 ( ) =. Assn (u) =, slt (w) L(w)\{ } n (v) L(v)\ ( {(u), (w)} w P 1 ((w)) ) ; t olorn xtns to P 1 n P. w P 10

11 Corollry.1. T onurtons (C17), (C18), n (C19) mt t tmplt (B1), n n ty r rul. Lmm.1. All onurtons mtn t tmplt (B) r rul. Proo. Lt (C, X, x) onurton mtn t tmplt (B) n lt L n (, )-lst ssnmnt. Lt (r) t unqu olor n t lst L(r). Lt L(u) = { 1, }. Sn P 1 s n xtr-spl pt, tr s t lst on {1, } su tt P u 1 ( ) =. Assn (u) =. I (r) / L(v), tn slt (w) L(w), n L(v) L(v) \ ( {(u), (w)} P w ((w)) ) ; t olorn xtns to P 1 n P. I (r) L(v), tn slt (w) L(w) \ L(v); osrv (w) (r). Tr xsts olor (v) L(v) \ ( {(r), (u)} P w ((w)) ) ; t olorn xtns to P 1 n P. Corollry.14. Usn Lmm., t onurtons (C0) n (C1) mt t tmplt (B), n n ty r rul. 4 No Cor 5-Cyl In ts ston w sow t s o orn or 5-yls rom Torm 1.7. Torm 4.1. I G s pln rp not ontnn or 5-yl, tn G s (4, )-oosl. Proo. Lt G ountrxmpl mnmzn n(g) mon ll pln rps von or 5- yls wt (4, )-lst ssnmnt L su tt G s not L-oosl. Osrv tt n(g) 4; n t, δ(g) 4. Sn G s mnml ountrxmpl, G os not ontn ny o t rul onurtons (C9) (C1). I (C, X, x) s rul onurton, 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 onurtons (C1) (C1) r lr nou tt w must onsr onurtons tt r orm y ntyn rtn prs o vrts n ts onurtons. In Appnx A, w onrtly k ll vrtx prs tt vo rtn or 5-yl n n tt ll rsultn onurtons r rul. For v V (G) n F (G) n ntl rs µ(v) = (v) 6 n ν() = l() 6. By Eulr s Formul, t sum o ntl rs s 1. Atr rs r ntlly ssn, t only lmnts wt ntv r r 4-vrts n 5-vrts. Sn or 5-yls r orn, tr s no -n n G n vry 4- s jnt to only 4 + -s. T possl rrnmnts o -, 4 + -, or 5 + -s nnt to 4- n 5-vrts r sown n Fur 6. Squntlly pply t ollown srn ruls. Not tt, or vrtx v n, w n µ (v) n ν () to t r on v n, rsptvly, tr pplyn rul (R). (R1) Lt v 4-vrtx n nnt to v. I s jnt to - tt s lso nnt to v, tn sns r 1 to v; otrws, sns r 1 to v. (R) Lt v 5-vrtx. I s nnt to v, tn sns r 1 to v. A s ny ν () < 0; otrws, s non-ny. (R) I v s 5-vrtx nnt to ny 5-, tn v sns r 1 to. 11

12 v v v v () () () () v v v v v () () () () () Fur 6: Possl yl rrnmnts o -, 4 + -, n 5 + -s nnt to 4- n 5-vrts A vrtx v s ny vrtx µ (v) < 0; otrws, v s non-ny. (R4) I s non-ny nnt to ny 5-vrtx v, tn sns r 1 to v. W sow tt µ 4 (v) 0 or vrtx v n ν 4 () 0 or. Sn t totl r ws prsrv urn t srn ruls, ts ontrts t ntv r sum rom t ntl r vlus. W n y onsrn t r struton tr pplyn (R1) n (R). Lt v vrtx. I v s 4-vrtx, tn µ(v) = n v rvs totl r t lst rom ts norn s y (R1). Furtrmor, v s not t y ny ruls tr (R1), so µ 4 (v) 0. I v s 6 + -vrtx, tn µ(v) 0 n v s not t y ny otr ruls, so µ 4 (v) 0. I v s 5-vrtx, tn µ(v) = 1 n v rvs totl r t lst 1 rom ts norn s y (R). Tror, or ny vrtx v, µ (v) 0. Lt. I s -, tn ν() = 0 n s not t y ny rul, so ν 4 () = 0. I s 4-, tn ν() =. In (R1) n (R), t only s tt sn r 1 to snl vrtx r jnt to -. A 4- jnt to - s or 5-yl, w s orn y ssumpton, so sns r t most 1 to vrtx. Sn 4-s r not t y ruls (R) (R4), ν 4 () 0. I s 6 + -, tn s t lst s mu ntl r s t s nnt vrts. I v s 4-vrtx nnt 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 s 5-vrtx nnt to, tn sns r 1 to v y (R1), n possly notr r 1 y (R4), n os not sn r to v y (R1) or (R). Tus sns r t most 1 to nnt vrtx, n ν 4 () 0. I s 5-, tn ν() = 4 n sns r t most 1 to nnt vrtx y (R1) n (R). Osrv tt ν () = 1, tn s nnt to v 4-vrts n s jnt to t lst on -; ts orms (C9), ontrton. Tror, w v t ollown lm out t strutur o ny 5-vrtx. Clm 4.. I s ny 5-, tn ν () = 1 n s jnt to xtly on 5-vrtx. W now onsr t r struton tr pplyn (R). I s ny 5-, tn ν () = 1 n s jnt to xtly on 5-vrtx, so ν () = 0. No s los r n (R), tror ν () 0 or ny. 1

13 Clm 4.. I v s ny 5-vrtx, tn v s nnt to tr -s, two 4 + -s, n xtly on ny 5-; n µ (v) = 1. Proo. Suppos tt v s vrtx su tt µ (v) < 0, n onsr t yl rrnmnt o - n 4 + -s out v. Cs 1: v s nnt to t lst our 4 + -s (Furs 6() n 6()). Sn µ (v) 1 n µ (v) < 0, v s nnt to t lst tr ny 5-s. Hn two o t ny 5-s r jnt, ormn (C1), ontrton. Cs : v s nnt to two non-jnt -s n tr 4 + -s (Fur 6()). Sn µ (v) = 1 n µ (v) < 0, v s nnt to two ny 5-s, 1 n. I ts two s r jnt, tn ty orm (C1), ontrton. Otrws, ty sr - t s nor n ll vrts nnt to 1,, n t otr tn v r 4-vrts, so t vrts nnt to 1 n t orm (C10), ontrton. Cs : v s nnt to two jnt -s n tr 4 + -s (Fur 6()). Sn µ (v) = 1 n µ (v) < 0, v s nnt to two ny 5-s, 1 n. I 1 n r jnt tn ty orm (C1), ontrton. Tus, 1 n r not jnt, ut ty r jnt to - nnt to v. Sn s ny or {1, }, snt r 1 to vry 4-vrtx nnt to. By (R1), vry 4-vrtx nnt to s nnt to - jnt to. Tror, 1 s jnt to - tt os not sr ny vrts wt t t two -s nnt to v, ormn on o (C0) or (C1), ontrton. Cs 4: v s nnt to tr -s n two 4 + -s (Fur 6()). I v s nnt to two ny 5-s 1 n, tn t - t jnt to ot 1 n s nnt to two 4-vrts, n t vrts nnt to 1 n t orm (C10), ontrton. Tror, v s nnt to xtly on ny 5-, s lm. By (R4), vry ny 5-vrtx rvs r 1 rom ts unqu nnt non-ny 5+ -, so µ 4 (v) 0 or vry vrtx v. E ny 5- s nonntv r tr (R), so ν 4 () < 0 or som 5-, tn sns r y (R4), n tus s non-ny. 1 t 1 t v 1 t v 5 v 4 v v +1 v +1 u t v +1 u t v +1 v w () A 5- wt ν 4() < 0. () Clm 4.5, Cs 1. () Clm 4.5, Cs. Fur 7: Spl ss or 5- wt ν 4 () < 0. 1

14 Consr t Fur 7(), wr s 5- wt ν 4 () < 0, s nnt to vrts v 1,..., v 5, v 1 s ny 5-vrtx, n 1 s t ny 5- nnt to v 1. Lt t 1 n t t jnt pr o -s nnt to v 1 wt t 1 jnt to 1 n t jnt to ; lt t t otr - nnt to v 1. W mk two s lms out ts rrnmnt. Clm 4.4. T vrtx v jnt to v 1 n nnt to t s 5 + -vrtx. Proo. I v s 4-vrtx, tn t vrts nnt to 1 n t orm (C10), ontrton. Clm 4.5. I v n v +1 r onsutv vrts on t orr o, tn t most on o v n v +1 s ny. Proo. Suppos tt two onsutv vrts v n v +1 r ny 5-vrts. Lt n +1 t ny 5-s nnt to v n v +1, rsptvly. Sn ot v n v +1 v tr nnt -s, s jnt to - t ross t v v +1. Lt u t tr vrtx nnt to t n onsr two ss. Cs 1: t s not n mon (Fur 7()). Sn s ny, t vrtx jnt to u n nnt to (wt v ) s 4-vrtx n s nnt to - t su tt t s jnt to. T vrts nnt to, +1, t, n t orm on o (C15) or (C19), ontrton. Cs : t s n mon (Fur 7()). Lt w t ourt vrtx n t mon n ssum, wtout loss o nrlty, tt v s jnt to w. Lt t vrtx nnt to +1 tt s not jnt to u or v +1 lon t ounry o +1 ; sn +1 s ny, tr s - t +1 nnt to n jnt to +1. T vrts v n w n tos nnt to +1 n t +1 orm on o (C17) or (C18), ontrton. By Clm 4.5, s nnt to t most two ny vrts, n y Clm 4.4, v s non-ny. I s nnt to xtly on ny 5-vrtx, tn v, v 4, n v 5 r 4-vrts sn µ () = 0, ut tn t vrts nnt to n 1 orm (C14), ontrton. Tror, s nnt to two ny vrts, n sn v s 5 + -vrtx, s nnt to xtly two 4-vrts. E o ts rvs r 1, so ν 4 () = 1. By Clm 4.5, t ny vrts nnt to onsst o v 1 n xtly on o v or v 4. T ny 5-vrtx v otr tn v 1 s lso nnt to tr -s t 4, t 5, n t 6, wr t 4 n t 5 orm mon wt t 4 jnt to. By Clm 4.4, t vrtx jnt to v n nnt to ot n t 6 s non-ny 5 + -vrtx. T only non-ny 5 + -vrtx nnt to s v, n n v s ny 5-vrtx n t 4 s nnt to v 4. I v s 6 + -vrtx, tn ν 4 () 0. Tror, tr s unqu rrnmnt o ny vrts, 4-vrts, n 5-vrtx out 5- wt ν 4 () < 0 (Fur 8). For {1, }, lt t ny 5- nnt to t ny 5-vrtx v. T vrts nnt to, 1,, t, n t 6 orm (C16), so ts rrnmnt os not ppr wtn G; n ν 4 () 0 or ll 5-s. Tror, vry vrtx n s nonntv r tr (R4), ontrtn t ntv ntl r sum. Tus, mnml ountrxmpl os not xst n vry pln rp wt no or 5-yl s (4, )-oosl. 14

15 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 nnt to non-ny 5- wt ν 4 () < 0. 5 No Cor 6-Cyl In ts ston w sow t s o orn or 6-yls rom Torm 1.7. T s o orn ouly-or 6- n 7-yls ollows rom vry smlr rumnt. W v t ull proo or no or 6-yls n sr t rns or t proo wn w or oulyor 6- n 7-yls. Torm 5.1. I G s pln rp not ontnn or 6-yl, tn G s (4, )-oosl. Proo. Lt G ountrxmpl mnmzn n(g) mon ll pln rps von or 6- yls wt (4, )-lst ssnmnt L su tt G s not L-oosl. Osrv tt n(g) 5; n t, δ(g) 4. Sn G s mnml ountrxmpl, G os not ontn ny o t rul onurtons. Splly, w us t t tt G vos (C) n (C4) (s Fur ). For v V (G) n F (G) n ntl r µ(v) = (v) 4 n ν() = l() 4. By Eulr s Formul, t ntl r sum s 8. Sn δ(g) 4, t only lmnts o ntv r r -s. Sn or 6-yl s orn n δ(g) 4, t lustrs (s Fur 1) r trnls (K), mons (K4), -ns (K5), 4-wls (K5), n 4-ns wt n vrts nt (K5). Splly not tt t 4-n (K6) ontns or 6-yl, so t most tr -s n lustr sr ommon vrtx, unlss ty orm 4-wl (K5) n t ommon vrtx s t 4-vrtx n t ntr o t wl. Apply t ollown srn ruls, s sown n Fur 9. (R1) I s - n s n nnt, tn lt t jnt to ross. (R1) I s 5 + -, tn pulls r 1 rom trou t. (R1) I s 4-, tn lt 1,, n t otr s nnt to. For {1,, }, lt t jnt to ross. For {1,, }, t pulls r 1 9 rom t trou t s n. (R) Lt v 5 + -vrtx, n lt n nnt -. (R) I v s 5-vrtx, tn v sns r 1 (R) I v s 6 + -vrtx, tn v sns r 4 9 to. to. (R) I X s lustr, tn vry - n X s ssn t vr r o ll -s n X. 15

16 (R1) (R1) (R) (R) v 1 v 4 9 Fur 9: Dsrn ruls n t proo o Torm 5.1. Not tt t ruls prsrv t sum o t rs. Lt µ (v) n ν () not t r on vrtx v or tr rul (R). W lm tt µ (v) 0 or vry vrtx v n ν () 0 or vry ; sn t totl r sum s prsrv y t srn ruls, ts ontrts t ntv r sum rom t ntl r vlus. Lt v vrtx. I v s 4-vrtx, tn v s not nvolv n ny rul, so t rsultn r s 0. I v s 6 + -vrtx, tn y (R) v loss r 4 9 to nnt -. Sn G vos or 6-yls, v s nnt to t most 4 (v) -s. Tus µ (v) stss µ (v) (v) (v) (v) (v) = (v) 4 0. I v s 5-vrtx, tn y (R) v loss r 1 6-yls, v s nnt to t most tr -s, so to nnt -. Sn G vos or µ (v) (v) 4 1 = (v) 5 = 0. Tror, µ (v) 0 or vry vrtx v. Lt. Sn 4-s r not jnt to 4-s, (R1) os not t t r vlu on 4-s. Tus, ν () = 0 or vry 4-. I s 6 + -, tn loss r t most 1 trou y (R1) or (R1), so ν () l() 4 1 l() = l() 4 0. Tror, ν () 0 or vry Lt 5-. Sn G ontns no or 6-yls, s not jnt to -. Tror, loss no r y (R1), ut oul los r usn (R1), so ν () l() l() = 8 l() Tror, ν () 0 s 5-. All ojts tt strt wt nonntv r v nonntv r tr t srn pross. It rmns to sow tt lustr o -s rvs nou r to rsult n nonntv r sum. 16

17 Cs 1: (K) Lt n solt -. T tr jnt s 1,, n r ll 4 + -s. By (R1) or (R1), rvs r 1 trou nnt, so ν () = = 0. Cs : (K4) Lt 1 n -s n mon lustr (K4). Tn 1 s jnt to two 4 + -s 1 n, n s jnt to two 4 + -s 1 n. By (R1) or (R1), t lustr rvs r 1 trou o t our s on t ounry o t mon. Sn ν( 1 ) + ν( ) =, t r vlu on t mon tr rul (R1) s. Sn G ontns no (C), tr s 5 + -vrtx v nnt to ot 1 n. I v s 5-vrtx, tn y (R), 1 n rv r 1, n t rsultn r on t mon s zro. I v s 6+ -vrtx, tn y (R), 1 n rv r 4 9, n t rsultn r on t mon s postv. Cs : (K5) Lt 1,, n -s n -n lustr (K5), wr s jnt to ot 1 n. T ntl r on ts lustr s. Tr r v s on t ounry o ts lustr, so y (R1) t lustr rvs r 5, rsultn n r 4 tr (R1). Not tt t s jnt to ot 1 n. Sn G ontns no (C), tr xsts 5 + -vrtx v nnt to ot 1 n, n tr xsts 5 + -vrtx u nnt 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 o n, rsultn n nonntv r on t -n. I v = u n v s 6 + -vrtx, tn y (R) v sns r 4 9 to 1,, n, rsultn n nonntv r on t -n. Otrws, suppos tt v = u n v s 5-vrtx. Sn G ontns no (C4), tr xsts notr 5 + -vrtx w nnt to t lst on o 1 n. By (R) v sns r 1 to o 1,, n, n y (R) w sns r t lst 1 to t lst on o 1 n, rsultn n nonntv r on t -n. Cs 4: (K5) Lt 1,,, n 4 -s n 4-wl (K5). T ntl r on ts lustr s 4. Tr r our s on t ounry o ts lustr, so y (R1) t lustr rvs r 4, rsultn n r 8 tr (R1). Lt v t 4-vrtx nnt to ll our -s. Lt u 1, u, u, n u 4 t vrts jnt to v, orr yllly su tt vu u +1 s t ounry o t - or {1,, } n vu 4 u 1 s t ounry o 4. Sn G ontns no (C) n (v) = 4, u s 5 + -vrtx. By (R), u sns r t lst to t lustr, rsultn n nonntv totl r. Cs 5: (K5) Lt 1,,, n 4 -s n 4-strp wt nt vrts s n (K5). T ntl r on ts lustr s 4. Lt v, u 1, u, u, n u 4 t vrts n t 4-strp, wr v s nnt to only 1 n 4, u 1 s nnt to only 1 n, u s nnt to,, n 4, u s nnt to 1,, n, n u 4 s nnt to only n 4. Tr r sx s on t ounry o ts lustr, so y (R1) t lustr rvs r 6, rsultn n r = tr (R1). 6 Sn n orm mon, n G ontns no (C), on o u n u s 5 + -vrtx. Wtout loss o nrlty, ssum u s 5 + -vrtx. Sn n 4 orm mon, n G ontns no (C), on o u n u 4 s 5 + -vrtx. I u s 5 + -vrtx, tn y (R), t lustr rvs r t lst + rom u n u, w rsults n nonntv totl r. Otrws, u s 4-vrtx n u 4 s 5 + -vrtx. I u s 6 + -vrtx, tn y (R), t lustr rvs r t lst 4 + rom u n u 4. I u s 5-vrtx, tn sn 1 n orm mon n G ontns no (C4), on o v n u 1 s 5 + -vrtx. By (R), t lustr rvs 17

18 r t lst + + rom u n u 4 n on o v n u 1. In tr s, t nl r s nonntv. W v vr tt t totl r tr srn s nonntv, ontrtn t ntv ntl r sum. Tus, mnml ountrxmpl os not xst n vry plnr rp wt no or 6-yl s (4, )-oosl. Corollry 5.. I G s pln rp not ontnn ouly-or 6-yl or ouly-or 7-yl, tn G s (4, )-oosl. Proo. Lt G mnml ountrxmpl y mnmzn n(g). Osrv tt n(g) 4 n δ(g) 4. Sn G ontns no ouly-or 6-yl, t lustrs r -s (K), mons (K4), -ns (K5), 4-wls (K5), n 4-ns wt n vrts nt (K5). Us t sm srn rumnt s n Torm 5.1, wt t ollown ns: I s 4-, tn n jnt to 4-. Howvr, sn G ontns no oulyor 7-yl, nnot jnt to -. Tror, os not los r y rul (R1). I s 5-, tn n jnt to t most on -, sn G ontns no oulyor 7-yl. By (R1) loss r 1 ross t t srs wt, n y (R1) loss r t most 1 9 ross t otr our s. Tus ν () l() = 9 0. All o t otr rumnts rom t proo o Torm 5.1 ol, w sows tt t rsultn totl r s nonntv, n n mnml ountrxmpl os not xst. 6 No Cor 7-Cyl Torm 6.1. I G s pln rp not ontnn or 7-yl, tn G s (4, )-oosl. W prov t ollown strntn sttmnt: Torm 6.. Lt G plnr rp wt no or 7-yl, n lt P surp o G, wr P s somorp to on o P 1, P, P, or K, n ll vrts n V (P ) r nnt to ommon. Lt L (4, )-lst ssnmnt o G P n lt propr olorn o P. Tr xsts n xtnson o to propr olorn o G su tt (v) L(v) or ll v V (G P ). Proo. Suppos tt tr xsts ountrxmpl. Slt ountrxmpl (G, P, L, ) y mnmzn n(g) 1 4n(P ) mon ll or 7-yl r pln rps, G, wt surp P somorp to rp n {P 1, P, P, K }, propr olorn o P, n (4, )-lst ssnmnt L o G P su tt os not xtn to n L-olorn o G. W wll rr to t vrts o P s prolor vrts. Clm 6.. G s -onnt. 18

19 Proo. I G s sonnt, tn onnt omponnt n olor sprtly. Suppos tt G s ut-vrtx v. Tn tr xst 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 wtout loss o nrlty tt G 1 ontns t lst on vrtx o P, so lt S 1 t surp o P ontn n G 1. I G ontns t lst on vrtx o P,.., v S 1, tn lt S t surp o P ontn n G ; otrws, lt S t vrtx v. Sn (G, P, L, ) s mnml ountrxmpl, tr s n L-olorn 1 o G 1 tt xtns t olorn on S 1. Usn t olor prsr y 1 on v, tr xsts n L-olorn o G tt xtns t olorn on S. T olorns 1 n orm n L-olorn o G, ontrton. Clm 6.4. G s no sprtn -yls. Proo. Suppos tt P = v 1 v v s sprtn -yl o G. Lt G 1 t surp o G vn y t xtror o P lon wt P, n lt G t surp o G vn y t ntror o P lon wt P. Sn P s sprtn, n(g 1 ) < n(g) n n(g ) < n(g). Sn t vrts n P sr ommon, w n ssum wtout loss o nrlty tt V (P ) V (G 1 ). Sn (G, P, L, ) s mnml ountrxmpl, tr xsts n L-olorn 1 o G 1. Assn t olors rom 1 to P. Tn tr xsts n L-olorn o G xtnn t olors on P, n totr 1 n orm n L-olorn o G, ontrton. Clm 6.5. I v V (P ) su tt V (P ) N[v], tn t surp o G nu y N(v) s not somorp to ny rp n {P 1, P, P, K }. Proo. Suppos tt tr xsts vrtx v V (P ) wr ll prolor vrts r n N[v] n t surp G[N(v)] s somorp to surp n {P 1, P, P, K }. Tn onsr t rp G = G v wt surp P = G[N(v)]. Sn N G [v] 4, tr xsts n L-olorn o G[N[v]]. Sn (G, P, L, ) s mnml ountrxmpl, xtns to n L-olorn o G, w n turn xtns to n L-olorn o G, ontrton. Clm 6.6. I v V (P ) s G (v), tn G (v) = n P s somorp 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 somorp to P, ontrtn Clm 6.5. Clm 6.7. P s somorp to on o P or K. Proo. Suppos tt P s not somorp to tr P or K. I P s somorp to P 1, tn t vrtx o P s two onsutv nors u 1 n u not n P ; lt U = {u 1, u }. I P s somorp to P, tn som vrtx v n P s nor u 1 not n P tt srs wt t n P ; lt U = {u 1 }. Lt P t surp somorp to P or K vn y nlun vrts n U. Tr xsts propr olorn o P tt xtns t olorn on P. But tn (G, P, L, ) s n(g) 1 4 n(p ) < n(g) 1 4n(P ), so tr xsts n L-olorn o G tt xtns, ontrton. Clm 6.8. I v V (G P ), tn G (v) 4. 19

20 Proo. Suppos tt v V (G P ) s r (v). Tn G v s plnr rp wt no or 7-yl ontnn prolor surp P n lst ssnmnt L. Sn (G, P, L, ) s mnmum ountrxmpl, G v s n L-olorn. Howvr, v s t most tr nors n t lst our olors n t lst L(v). Tus, tr s n xtnson o t L-olorn o G v to n L-olorn o G, ontrton. Osrv tt n(g) 4. Rll tt n onurton (C, X, x), n L-olorn o V (C) \ X xtns to ll o C. Bus o ts t, G ontns rul onurton (C, X, x), tn tr s prolor vrtx n t st X, or ls G X s n L-olorn tt xtns to ll o G. Splly, w wll us t t tt G vos (C), (C), (C4), (C5), (C6), (C7), n (C8). For 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 ounry o s t st o prolor vrts, V (P ). By Eulr s Formul, t ntl r sum s t most 1. Clms 6.6 n 6.8 ssrt tt t only ntvly-r ojts r -s. For vrtx v, lt t k (v) not t numr o k-s nnt to v. Apply t ollown srn ruls. Lt µ (v) n ν () not t r on vrtx v or tr rul (R). 8 v 1 v 4 9 (R1) (R) (R) (R1) (R1), Cs 1 (R1), Cs Fur 10: Dsrn ruls (R1) n (R) n t proo o Torm 6.1. (R0) I v s prolor vrtx n s n nnt - wt ntv r, tn v sns r 1 to. (R1) I s - n s n nnt, tn lt t jnt to ross. 0

21 (R1) I s 5 + -, tn pulls r 8 rom trou t. (R1) I s 4- n s t only - jnt to, tn lt 1,, n t otr s nnt to. For {1,, }, lt t jnt to ross. For {1,, }, t pulls r 1 8 rom t trou t s n. (R1) I s 4- n s jnt to two -s 1 n (sy 1 = ), tn lt 1 n t otr s nnt to, wr t s 1 n srn ts s r 6 + -s. For {1, }, t pulls r 16 rom t trou t s n. (R) Lt v 5 + -vrtx wt v / V (P ) n lt n nnt -. (R) I v s 5-vrtx, tn v sns r 1 to, wn = mx{, t (v)}. (R) I v s 6 + -vrtx, tn v sns r 1 (R) I s 6- wt ν () < 0 n v s n nnt 5 + -vrtx or n nnt vrtx n V (P ) wt µ 0 (v) > 0, tn v sns r 1 4 to. W lm tt µ (v) 0 or vry vrtx v n ν () 0 or vry. Sn t totl r sum ws prsrv urn t srn ruls, ts ontrts t ntv r sum rom t ntl r vlus. Not tt 6-s r not nnt to -s sn G os not ontn or 7-yl. Osrv tt 6- s ν 1 () < 0 n only ll s jnt to r 4-s, n o tos 4-s s two jnt -s. Clm 6.9. Lt v vrtx n V (P ). Tn µ (v) 0. In ton, v s nnt to 6- 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 r µ (v) s nonntv. Sn (v) t (v) + t 6 (v), t sus to sow tt µ 0 (v) 1 4 (v) t (v). Cs 1: P = P. Lt v 1, v, n v t vrts n t -pt P. For {1,, }, µ(v ) = (v ). Sn P s not somorp to K, ts vrts o not orm yl, n t to w ll vrts r nnt s not -. Hn t (v ) (v ) 1. I (v ) 4, tn µ(v ) = (v ) 1 (v ) > 1 4 (v ) t (v ). I (v ) =, tn µ(v ) = 0. I =, tn v s not nnt to ny -s sn v 1 n v r not jnt. I {1, } n v s jnt to -, tn lt v t nor o v not n V (P ). Lt P t surp nu y (V (P ) {v }) \ {v }, w orms opy o P or K n G v. For ny olor (v ) L(v ) \ {(v )}, tr xsts n L-olorn o G v s (G v, P, L, ) s not ountrxmpl; ts olorn xtns to n L-olorn o G. Tus, t (v ) = 0. I v s nnt to 6- wt ν 1 () < 0, tn t otr nnt to v s 4- tt s jnt to two -s. Ts rsults n or 7-yl, ontrton; tus (R) os not pply to v. to. I (v ) =, Clm 6.4 ssrts tt G s no sprtn -yls, so tn v loss r t most 1 n (R0). I v s nnt to 6- wt ν 1 () < 0, tn t otr two s nnt to v 1

22 r 4-s n ts 4-s r jnt to two -s. Ts rts or 7-yl, ontrton, so (R) os not pply to v n µ (v ) 0. Cs : P = K. Lt v 1, v, n v t vrts n t -yl P, so µ(v ) = (v ) or v. By Clm 6.4, G s no sprtn -yl, so t tr vrts r nnt to ommon - wt ν() = 0. Tror, vrtx v sns r 1 to t most (v ) 1 nnt -s y (R0). Rll tt (v ) y Clm 6.6. Suppos tt (v ) =. I t (v ) > 1, t surp o G nu y t noroo o v s somorp to P or K, ontrtn Clm 6.5. I (v ) 4, tn µ(v ) = (v ) 1 (v ) 1 4 (v ) t (v ). Tror, µ (v ) 0. Tus, n ll ss prolor vrtx v s µ (v) 0. W wll now sow tt ll ojts tt strt wt nonntv r lso n wt nonntv r. I s 4-, tn (R1) n (R1) o not pull r rom, sn ts woul rqur to jnt to 4- tt s jnt to - t, ut tn,, n t orm ouly-or 7-yl. Tus, ν () = 0 or vry 4-. I s 5-, tn sn G ontns no or 7-yls, s not jnt to two -s n s not jnt to 4-. Tror, loss r t most 8 y (R1), ut loss no r usn (R1), so ν () > 0 or vry 5-. I s 6-, tn s not jnt to - sn G ontns no or 7-yl. Osrv tt y Clm 6. t ounry o s smpl 6-yl. So sns r trou n urn (R1), t n sn r 1 8 trou y (R1), or t n sn r 8 trou y (R1). T only wy tt ts wll rsult n ntv r tr (R1) n (R) s or to sn r 8 trou o ts sx s y (R1); ts wll us ν () = 6 8 = 1 4. I s prolor vrtx v on ts ounry, tn y Clm 6.9, v s postv r tr (R0); y (R), rvs r t lst 1 4, rsultn n ν () 0. I s no nnt prolor vrts, tn sn G ontns no (C), som vrtx v on t ounry o s 5 + -vrtx. By (R) v sns r 1 4 to n n ν () 0. Osrv t ollown lm onrnn t strutur out vrtx tt loss r y (R). Clm Lt v 5 + -vrtx wt t tr nnt s 1,, n, n yl orr. I v sns r to y (R), tn 1 n r 4-s n s 6-. I s 7 + -, tn y (R1) loss r t most 8 ν () l() 4 8 l() = 5 l() 4 > 0. 8 trou. Tus, Tror, ν () > 0 or vry Nxt, w wll onsr vrtx v not n V (P ). I v s 4-vrtx, tn v os not los r y ny rul, so t rsultn r s 0. I v s 5-vrtx, lt = mx{, t (v)} n v loss r 1 t (v) to nnt -s y (R). I (R) os not pply to v, tn v sns r t most 1 to nnt -s n µ (v) 0. I (R) ppls to v, tn v s nnt to s 1,, n wr 1 n r 4-s n s 6-. Sn (v) = 5 n G s no or 7-yl, t rul (R) ppls t most on. I (R) ppls on, tn t (v) n v loss r t most y (R) n r 1 4 y (R), so µ (v) 0. I v s 6 + -vrtx, tn lt k = t (v) n l t numr o tms (R) ppls to v. Not tt k 4 5(v) sn G vos or 7-yls. Furtr, not tt k + l (v), sn

23 6- tt ns r rom v y (R) s pr y 4- n t yl orr o s roun v. By (R), v n los r 1 to nnt -, n v n los r t most 1 4 to nnt 6- y (R). Tn v ns wt 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 stsy t ollown lnr prorm wt ul on vrls 1,, n : mn 1 k 1 4 l s.t k 0 k l 0, k, l 0 mx 7 1 s.t ,, 0 T ul-sl soluton ( 1,, ) = ( 40, 1 0, 1 ) 4 monstrts tt 1 k 1 4 l 7 40 > 4, n tus µ (v) > 0 or vry 7 + -vrtx. It rmns to sown tt t lustrs rv nou r to om nonntv. Sn G ontns no sprtn -yl, G os not ontn t lustr (K5) or t lustrs (K6) (K6r). Osrv tt tr s no prolor vrtx v o r t most tr wr ll s nnt to v v lnt tr. Fnlly, t s wort notn n tt G ontns rul onurton (C, X, x), tn tr s prolor 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 r 1 to nnt ntvly-r - y (R0) or (R). I P = K, w ll P t prolor. v 1 1 (K) (K4) (K5) Fur 11: Clustrs (K), (K4), n (K5) Cs 1: (K) Lt t solt - n (K). I s t prolor, tn ν () = ν() = 0. Otrws, t ntl r on s 1. By (R1), rvs r 9 8 trou ts ounry s, rsultn n nonntv nl r. Cs : (K4) Lt 1 n -s n mon lustr (K4). Frst, suppos wtout loss o nrlty tt 1 s t prolor. T ntl r o t lustr s 1. Tn rvs r 1 y (R0) n r 8 y (R1), rsultn n postv nl r. Otrws, t ntl r on t lustr s. By (R1), 1 n rv r 8 trou o t two s on t ounry o t lustr, rsultn n r 1. I t lustr ontns prolor vrtx u, tn t rvs r 1 y (R0). Otrws, sn G ontns no (C), tr s 5+ -vrtx v nnt to ot 1 n. By (R), ts vrtx sns r t lst 1 to o t s, rsultn n nonntv nl r. Cs : (K5) Lt 1,, n -s n -n lustr (K5), wr s jnt to ot 1 n. Suppos tt t lustr ontns prolor, so t ntl r on t lustr

24 s. I s prolor, tn t lustr rvs r 4 1 y (R0); 1 or s prolor, tn t lustr rvs r 1 y (R0) n r 8 y (R1). In tr s, t nl r s nonntv. I P = K or t lustr os not ontn t prolor, tn t ntl r on t lustr s. By (R1), t lustr rvs r 5 8, rsultn n r 9 8. Not tt t s 1 n orm mon n t s n orm mon. Sn G ontns no (C), tr xsts ull vrtx v nnt to ot 1 n. Smlrly, tr xsts ull vrtx u nnt to n. I u v, tn y (R0) or (R), v sns r t lst 1 to o 1 n n u sns r t lst 1 to o n, rsultn n nonntv r on t lustr. I u = v n v s vy vrtx, tn v sns r 1 to 1,, n, rsultn n nonntv r on t lustr. Otrws, suppos tt u = v / V (P ) n v s 5-vrtx. Sn G ontns no (C4), tr xsts notr ull vrtx w tt s nnt to t lst on o 1 n. By (R), v sns r 1 to 1,, n, n y (R0) or (R), w sns r t lst 1 to on o 1 n, rsultn n nonntv r on t lustr. u 1 u 1 v u 4 v u 5 u 4 u u 1 u 1 (K5) (K6) (K6) Fur 1: Clustrs (K5), (K6), n (K6) Cs 4: (K5) Lt 1,,, n 4 -s n 4-wl (K5). I t lustr ontns prolor, tn t ntl r on t lustr s ; t lustr rvs r 5 1 y (R0) n r 8 y (R1), rsultn n postv nl r. Otrws, t ntl r on ts lustr s 4. By (R1), t lustr rvs r 4 8, rsultn n r 5. Lt v t 4-vrtx nnt to ll our -s. Lt u 1, u, u, n u 4 t vrts jnt to v, orr yllly su tt vu u +1 s t ounry o t - or {1,, } n vu 4 u 1 s t ounry o 4. Sn t lustr os not ontn t prolor, v s not prolor vrtx. Sn G ontns no (C), u s ull vrtx. Wn u s 5-vrtx, t s nnt to two 7 + -s, so u sns r 1 to nnt - y (R). Tus, u sns r t lst 1 to t lustr y (R0) or (R), rsultn n nonntv nl r. Cs 5: (K6) Lt 1,,, n 4 -s n 4-strp lustr (K6). I t lustr ontns t prolor, tn t ntl r on t lustr s. I 1 or 4 s prolor, tn t lustr rvs r 1 y (R0) n r 4 8 y (R1); or s prolor, tn t lustr rvs r 5 1 y (R0) n r 5 8 y (R1). In tr s, t rsultn nl r s nonntv. I t lustr os not ontn t prolor, tn t ntl r on ts lustr s 4. By (R1), t lustr rvs r 6 8, rsultn n r 7 4. Not tt or {1,, }, t s n +1 orm mon. Sn G ontns no (C), tr xsts ull vrtx v nnt to ot n +1. Lt u 1 ull vrtx nnt to n. Wtout loss o nrlty, u 1 s not nnt to 4, so tr s ull vrtx u nnt to 1 n. I u 1 s vy vrtx, t lustr rvs r 1 rom u 1 y (R0) or (R), n r t lst 1 rom u y (R0) or (R), rsultn n postv nl r. Otrws, 4 u 4

25 u 1 s 5-vrtx, so u 1 sns r 1 y (R), rsultn n r 4. I u s nnt to, tn u sns r t lst 1 y (R0) or (R), rsultn n postv nl r. Otrws, u s nnt wt 1 n ut not. I u s lr vrtx, t sns r 1 y (R0) or (R). Otrws, sn G ontns ntr (C) or (C4), tr s tr ull vrtx u. T lustr rvs r 1 rom u y (R) n r t lst 1 rom u y (R0) or (R). In s, t rsultn nl r s nonntv. Cs 6: (K6) Lt 1,,, n 4 -s n 4-n lustr (K6). Lt v t ntr o t n, wt nors u 1, u, u, u 4, n u 5 wr or {1,, }, n +1 r jnt on t vu +1. I t lustr ontns t prolor, tn t ntl r on t lustr s. I 1 or 4 s prolor, tn t lustr rvs r 4 1 y (R0) n r 4 8 y (R1); or s prolor, tn t lustr rvs r 5 1 y (R0) n r 5 8 y (R1). In tr s, t rsultn nl r s postv. I t lustr os not ontn t prolor, tn t ntl r on ts lustr s 4. By (R1), t lustr rvs r 6 8, rsultn n r 7 4. I v s vy vrtx, tn y (R0) or (R) v sns r 4 1 to t lustr, rsultn n postv r. Otrws, v / V (P ) n v s 5-vrtx, so v sns r 1 to t lustr y (R), rsultn n r 4. I tr s vy vrtx n {u, u, u 4 }, tn tt vrtx ontruts r 1 to t lustr, rsultn n postv r. I tr s no vy vrtx n {u, u, u 4 }, tn tr s t lst on 5-vrtx n {u, u, u 4 } sn G ontns no (C4). I tr r multpl 5-vrts n {u, u, u 4 }, tn sns r 1 to t lustr y (R), rsultn n postv r. I tr s only 5-vrtx w mon u, u, n u 4, tn tr s ull vrtx z {u 1, u 5 } sn G os not ontn (C4) or (C5); t lustr rvs r 1 rom w y (R) n t lst 1 rom z y (R0) or (R), rsultn n postv nl r. u 1 u u 1 u 6 u 4 u 5 u 4 u (K6) v 4 1 u 1 u 4 (K6) w Fur 1: Clustrs (K6) n (K6). Cs 7: (K6) Lt 1,,, n 4 t -s o ts lustr (K6) wr 4 s jnt to or {1,, }. I t lustr ontns t prolor, tn t ntl r on t lustr s. I on o 1, or s prolor, t lustr rvs r 4 1 y (R0) n r 4 8 y (R1). I 4 s prolor, tn t lustr rvs r 6 1 y (R0). In tr s, t rsultn nl r s nonntv. I t lustr os not ontn t prolor, tn t ntl r on t lustr s 4. By (R1), t lustr rvs r 6 8, rsultn n r 7 4. Lt u 1, u, u, u 4, u 5, n u 6 t vrts on t ounry o t lustr orr su tt u, u 4, u 6 r t vrts nnt to 1 n, n, n n 1, rsptvly. Sn G ontns no (C), tr r t lst two ull vrts n {u, u 4, u 6 }. By (R0) or (R), ts vrts sn r t lst 1 to t lustr, rsultn n postv totl r. 5

26 Cs 8: (K6) Lt 1,,, n 4 yllly-orr -s n 4-wl wt ntr vrtx v wr n +1 sr ommon or {1,,, 4}, wr ns r tkn moulo 4; lt - jnt to 4 ut not nnt to v, ompltn our lustr (K6). I t lustr ontns t prolor, tn t ntl r on t lustr s 4. I 1 or s prolor, tn t lustr rvs r 6 1 y (R0) n r 4 8 y (R1). I s prolor, tn t lustr rvs r 5 1 y (R0) n r 4 8 y (R1). I 4 s prolor, tn t lustr rvs r 7 1 y (R0) n r 5 8 y (R1). In o t ov ss, t nl r s nonntv. I s prolor, tn t lustr rvs r 4 1 y (R0) n r 8 y (R1), rsultn n r 7 8. Lt N(v) = {u 1, u, u, u 4 } wr u s nnt to n +1 or ll {1,,, 4}. Sn G os not ontn (C), u 1 n u r ull vrts. E o u 1 n u sns r t lst 1 to t lustr y (R), rsultn n nonntv r. I t lustr os not ontn t prolor, tn t ntl r on ts lustr s 5 n v / V (P ). By (R1), t lustr rvs r 5 8, rsultn n r 5 8. Sn G os not ontn (C), u 1, u, u, n u 4 r ull vrts. By (R0) or (R), t lustr rvs r t lst 1 rom o u 1 n u n r t lst 1 rom o u n u 4, rsultn n postv nl r. 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 t yllly-orr -s n 5-wl wt ntr vrtx v wr n +1 sr ommon or {1,,, 4, 5}, wr ns r tkn moulo 5. Lt N(v) = {u 1, u, u, u 4, u 5 } wr u s nnt to n +1 or {1,,, 4, 5}. I t lustr ontns t prolor, tn t ntl r on t lustr s 4. T lustr rvs r 6 1 y (R0) n r 4 8 y (R1), rsultn n postv nl r. I t lustr os not ontn t prolor, tn t ntl r s 5 n v / V (P ). By (R1), t lustr rvs r 5 8, n y (R), t lustr rvs r 1 rom v, rsultn n r Sn G os not ontn (C4) or (C6), tr r t lst tr ull vrts n N(v). I N(v) ontns t lst tr vy vrts, tn t lustr rvs r t lst 6 1 y (R0) or (R), rsultn n postv nl r. I N(v) ontns xtly two vy vrts, tn t lustr rvs r y (R0) or (R) n r rom ull vrtx y (R), rsultn n postv r. I N(v) ontns xtly on vy vrtx, tn t lustr rvs r 1 1 y (R0) or (R) n r rom o two ull vrts y (R), rsultn n postv nl r. I N(v) ontns no vy vrts, tn tr r t lst tr ull vrts n N(v). Sn G os not ontn (C4), tr r t lst two nonjnt 5-vrts n N(v). Furtr, sn G os not ontn (C6), (C7), or (C8), tr r t lst our 5-vrts n N(v). T lustr 6

27 rvs r 1 rom o ts vrts y (R), rsultn n postv nl r. Cs 10: (K6) Lt 1 n t ntror -s n t two ovrlppn 4-wls tt mk up t lustr (K6). Lt u 1 n u t sr vrts o 1 n n lt z n w t vrts nnt wt 1 n, rsptvly, tt v not yt n ll. Sn G ontns no (C), t lst on o u 1 n u s n V (P ). Tn sn ll t prolor vrts l on ommon, t lustr ontns t prolor, so t ntl r s 5. I 1 or s prolor, tn t lustr rvs r 8 1 y (R0) n r 4 8 y (R1), rsultn n postv nl r. I on o t otr -s s prolor, tn t lustr rvs r 6 1 y (R0) n r 8 y (R1), rsultn n r 7 8. Sn G ontns no (C), on o w n z s non-prolor 5 + -vrtx. Ts vrtx sns r t lst 1 to t lustr y (R), rsultn n postv nl r. W v vr tt t totl r tr srn s nonntv, ontrtn t ntv ntl r sum. Tus, mnml ountrxmpl os not xst n vry plnr rp wt no or 7-yl s (4, )-oosl. 7 Conluson n Futur Work W prov tt, or k {5, 6, 7}, plnr rps wt no or k-yls r (4, )-oosl. Our mtos or provn rul onurtons rt svrl lr lsss o rul onurtons, su s tmplts; nturlly, tr r mny mor rul onurtons tn t ons w xpltly us. On oul lkly prov tt G s plnr rp wt no or 4-yl n no ouly-or 7-yl, tn G s (4,)-oosl usn mtos smlr to tos n ts ppr. W wr unl to xtn ts rsults to prov Conjtur 1., tt ll plnr rps r (4, )-oosl. Aknowlmnts W tnk Ryn R. Mrtn, Alx Nowk, Alx Sult, n Sns Wlkr or prtpton n t rly sts o t projt. Rrns [1] N. Alon n M. Trs. Colorns n ornttons o rps. Comntor, 1():15 14, 199. [] O. V. Boron. Colorns o pln rps: survy. Dsrt Mtmts, 1(4):517 59, 01. [] O. V. Boron n A. O. Ivnov. Plnr rps wtout trnulr 4-yls r 4-oosl. S. Élktron. Mt. Izv., 5:75 79, 008. [4] I. Co, B. Lký, n D. Stol. On ooslty wt sprton o plnr rps wt orn yls. Journl o Grp Tory. to ppr. [5] D. Crnston n D. B. Wst. A u to t srn mto. rxv: [mt.co]. [6] B. Frz. Plnr rps wtout 7-yls r 4-oosl. SIAM Journl o Dsrt Mtmts, (): ,

28 [7] G. Fjvz, M. Juvn, B. Mor, n R. Škrkovsk. Plnr rps wtout yls o sp lnts. Europn Journl o Comntors, (4):77 88, 00. [8] H. Krst n B. Lký. On ooslty wt sprton o plnr rps wt lsts o rnt szs. Dsrt Mtmts, 9(10): , 015. [9] J. Krtovíl, Z. Tuz, n M. Vot. Brooks-typ torms or ooslty wt sprton. Journl o Grp Tory, 7(1):4 49, [10] P. Lm, B. Xu, n J. Lu. T 4-ooslty o pln rps wtout 4-yls. Journl o Comntorl Tory, Srs B, 76(1):117 16, [11] C. Tomssn. Evry plnr rp s 5-oosl. Journl o Comntorl Tory, Srs B, 6(1): , [1] M. Vot. Lst olourns o plnr rps. Dsrt Mtmts, 10(1):15 19, 199. [1] R. Škrkovsk. A not on ooslty wt sprton or plnr rps. Ars Comntor, 58: , 001. [14] W. Wn n K. L. Cooslty n ooslty o plnr rps wtout v yls. Appl Mtmts Lttrs, 15(5): , 00. [15] D. B. Wst. Introuton to rp tory, Son ton. Prnt Hll, 001. A Lr Rul Conurtons In t proo o Torm 4.1, w monstrt tt no mnml ountrxmpl xsts y sown tt tr xsts rul onurton (C, X, x) wr G ontns opy o C[X] s n nu surp (n lso t opy rs wt t xtrnl rs). In ts ppnx, w prov t tls tt lry ts ssumpton. By Lmm., w n rlx t onton tt C[X] s n nu surp. W wll monstrt tt t onurtons tt ppr tr som vrts n X r mr (wl lso prsrvn t lnts, vrtx rs, n lk o or 5-yl) rsult n rul onurtons. Lt (C, X, x) rul onurton n lt {x 1, x 1},..., {x t, x t} lst o vrtx prs n X. For ts onurtons, w my nty som -yls n 5-yls tt r rqur to 5-s (n t ontxt o t proo o Torm 4.1). T rsultn onurton (C, X, x) wr C n X r mo rom C n X y mrn x wt x n rmovn ny mults or loops tt rsult. W sy lst {x 1, x 1},..., {x t, x t} s vl or (C, X, x) t rsultn onurton (C, X, x) my ppr n plnr rp o mnmum r t lst our ontnn no or 5-yl. Tr r tr stutons tt n our wn w prorm ts ton. Prs too los: I som pr {x, x } v (x, x ), tn tr w rt loop or mult wn mrn x n x. Ts wll ru t r o t rsultn vrtx, n ton to possly sortnn known - n 5-yls. Sn stns only rs s vrts r mr, pr ln ts proprty wll not ppr n ny vl lst o prs. Prs rtn or: I mrn x n x rts or 5-yl, tn ts onurton woul not ppr n t mnml ountrxmpl rom Torm 4.1. Sn stns only rs s vrts r mr, pr ln ts proprty wll not ppr n ny vl lst o prs. Rul prs: I mrn x n x os not t n t ov two ss, tn w wll monstrt tt t rsultn onurton s rul. Evn mrn on pr o vrts rts rul onurton, w n to k ll possl lsts o prs tt ontn tt pr. Atr onsrn ll prs tt oul nt, osrv tt n s tr s no st o tr or mor vrts wr vry pr n nt. In t ollown tls, w lst on o t onurtons (C10) (C1), ll t vrts, n lst ll prs o vrts nto t tr tors ov. In t s o rul prs, w prsnt t ontrt rp. Most o ts ontrt rps ontn opy o (C1), (C), (C10), (C11), or (C1). T only xptons 8

29 r t ontrt rps rv rom (C16), ut o ts onurtons s n Alon-Trs orntton n n s rul. (C10) Prs too los:,,,,,,,,,,,,,. Prs rtn or: Rul prs: Non rmn. (C11) Prs too los:,,,,,,,,,,,,,,,,,,. Prs rtn or:, Rul prs: Non rmn. (C1) Prs too los:,,,,,,,,,,,,,,. Prs rtn or: Non rmn. Rul prs: Non rmn. (C1) Prs too los:,,,,,,,,,,,,,,,,,. Prs rtn or:,,,,.. Rul prs: (ontns (C1)) Contns (C1) on 4-yl,,,. 9

30 (C14) Prs too los:,,,,,,,,,,,,,,,,,,,,,,,,,. Prs rtn or:,,,,,. Rul prs: (ontns (C11)), (ontns (C11)), n (ontns (C1)). Contns (C11) Contns (C11) Contns (C1) tr ltn vrtx. tr ltn vrtx. tr ltn vrtx. (C15) Prs too los:,,,,,,,,,,,,,,,,,,,,,,,,,,,. Prs rtn or:,,, (,,,,, ),, (,,,,, ),,. Rul prs: (ontns (C)), (ontns (C1)), n (ontns (C1)). Contns (C) Contns (C1) Contns (C1) on 6-yl,,,,,. on 4-yl,,,. on 4-yl,,,. 0

31 (C16) k j l m k j l m Prs too los:,,,,,,, j, k, m,,,,,j, k, l, m,,, j, m,,,,,,,, j,,,, j,,,, jk, jl, jm, kl, km, lm. Prs rtn or:, l,,,,, k,,, l,,, j, k, m, k, l, m, k, l, m, j,k, m, j, j, k, m. Rul prs: (s Alon-Trs orntton), l (symmtr to ), k (s Alon-Trs orntton), m (s Alon-Trs orntton), l (s Alon-Trs orntton), l (symmtr to k), l (symmtr to m). l m j k j k j k m l l m (C17) Prs too los:,,,,,,,,,,,,,,,,,,,,. Prs rtn or:,,,,,. Rul prs: Non rmnn. 1