c 2009 Society for Industrial and Applied Mathematics

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1 SIAM J. DISCRETE MATH. Vol. 0, No. 0, pp Soity or Inustril n Appli Mtmtis THE TWO-COLORING NUMBER AND DEGENERATE COLORINGS OF PLANAR GRAPHS HAL KIERSTEAD, BOJAN MOHAR, SIMON ŠPACAPAN, DAQING YANG, AND XUDING ZHU Astrt. T two-olorin numr o rps, wi ws oriinlly introu in t stuy o t m romti numr, lso ivs n uppr oun on t nrt romti numr s introu y Boroin. It is prov tt t two-olorin numr o ny plnr rp is t most nin. As onsqun, t nrt list romti numr o ny plnr rp is t most nin. It is lso sown tt t nrt ionl romti numr is t most 11 n t nrt ionl list romti numr is t most 12 or ll plnr rps. Ky wors. two-olorin numr, nrt olorin, plnr rp AMS sujt lssiition. 05C15 DOI / Introution. T two-olorin numr o rp ws introu y Cn n Slp [6] in t stuy o Rmsy proprtis o rps n us ltr in t stuy o t m romti numr [9, 10, 15, 16]. In [10] it ws sown tt t two-olorin numr is rlt to t yli romti numr. It turns out tt t two-olorin numr is lso rlt to t notion o nrt olorins introu y Boroin [1], wi ws t strtin motivtion or t rsults o tis ppr. Lt G rp, n lt L linr orrin o V (G). A vrtx x is k-rl rom y i x< L y n tr is n xy-pt P o lnt t most k su tt y< L z or ll intrior vrtis z o P.LtR L,k (y) t st o ll vrtis x tt r k-rl rom y wit rspt to t linr orr L. Tk-olorin numr o G is in s ol k (G) =1+min L mx R L,k(y), y V (G) wr t minimum is tkn ovr ll linr orrins L o V (G). I k =1,tn ol 1 (G) islsoknownstolorin numr o G sin it provis n uppr oun Riv y t itors Sptmr 25, 2007; pt or pulition (in rvis orm) July 7, 2009; pulis ltronilly DATE. ttp:// Dprtmnt o Mtmtis n Sttistis, Arizon Stt Univrsity, Tmp, AZ (kirst@su.u). Tis utor s rsr ws support in prt y NSA rnt H Dprtmnt o Mtmtis, Simon Frsr Univrsity, Burny, BC V5A 1S6, Cn (mor@su.). On lv rom Dprtmnt o Mtmtis, IMFM & FMF, Univrsity o Ljuljn, Ljuljn, Slovni. Tis utor s rsr ws support in prt y t ARRS (Slovni), Rsr Prorm P1 0297, y n NSERC Disovry rnt, n y t Cn Rsr Cir prorm. Univrsity o Mrior, FME, Smtnov 17, 2000 Mrior, Slovni (simon.sppn@unim.si). Tis utor s rsr ws support in prt y t ARRS (Slovni), Rsr Prorm P Cntr or Disrt Mtmtis, Fuzou Univrsity, Fuzou, Fujin , Cin (qin85@ yoo.om). Tis utor s rsr ws support in prt y NSFC unr rnts n SX o t oll o Fujin. Dprtmnt o Appli Mtmtis, Ntionl Sun Yt-sn Univrsity, Kosiun 80424, Tiwn, Popl s Rpuli o Cin, n Ntionl Cntr or Tortil Sins, Ntionl Tiwn Univrsity, Tipi 10617, Tiwn, Popl s Rpuli o Cin (zu@mt.nsysu.u.tw). Tis utor s rsr ws support y Tiwn rnt NSC M MY3. 1

2 2 KIERSTEAD, MOHAR, ŠPACAPAN, YANG, AND ZHU or t romti numr o G. T ir k-olorin numrs provi uppr ouns or som otr olorin prmtrs [17]. Lt k positiv intr. A rp G is k-nrt i vry surp o G s vrtx o r lss tn k. A olorin o rp su tt or vry k 1, t union o ny k olor lsss inus k-nrt surp, is nrt olorin. Not tt tis strntns t notion o yli olorins, or wi it is rquir tt vry olor lss is 1-nrt n t union o ny two olor lsss inus 2-nrt rp ( orst). T nrt romti numr o G, not s χ (G), is t lst n su tt G mits nrt olorin wit n olors. Suppos tt or vrtx v V (G) w ssin list L(v) N o missil olors wi n us to olor t vrtx v. A list olorin o G is untion : V (G) N, su tt (v) L(v) orv V (G) n(u) (v) wnvr u n v r jnt in G. I t olorin is lso nrt, w sy tt is nrt list olorin. I or ny oi o lists L(v),v V (G), su tt L(v) k, trxists list olorin o G, tn w sy tt G is k-oosl. T list romti numr (or t oi numr) og, not s (G), is t lst k, su tt G is k-oosl. Anloously w in t nrt oi numr n not it y (G). In 1976 Boroin prov [2, 3] tt vry plnr rp mits n yli 5-olorin n tus solv onjtur propos y Grünum [7]. At t sm tim, propos t ollowin onjtur. Conjtur 1 (Boroin [2, 3]). Evry plnr rp s 5-olorin su tt t union o vry k olor lsss wit 1 k 4 inus k-nrt rp. Tomssn sttl wknin o Conjtur 1 y provin tt t vrtx st o vry plnr rp n ompos into two sts tt, rsptivly, inu 2-nrt rp n 3-nrt rp [13], n tt t vrtx st o vry plnr rp n ompos into n inpnnt st n st tt inus 4-nrt rp [14]. Howvr, Conjtur 1 rmin silly untou sin tr r no tools to l wit nrt olorins. Vry rntly, t rrir ws ovrrin y Rutn [11] wo prov tt vry plnr rp mits nrt olorin usin t most 18 olors. In tis ppr w introu two irnt ppros or lin wit nrt olorins. Bot r s on t ollowin osrvtion. Osrvtion 1. Lt G rp, n lt nrt olorin o vrtxlt surp G v. I t niors o v r olor y pirwis istint olors n w olor v y olor wi is irnt rom ll o tos olors, tn t rsultin olorin o G is nrt. T osrvtion ov, wos sy proo is lt to t rr, is link twn t two-olorin numr n t nrt romti numr, sin it implis t ollowin proposition. Proposition 1. For ny rp G, (G) ol 2 (G). Proo. Suppos tt L is linr orrin o V (G), n suppos tt vrtx y s list o R L,2 (y) + 1 olors. Tn w n olor t vrtis o G, onyon, orin to t linr orrin L, sottvrtxy is olor y olor irnt rom t olors o vrtis in R L,2 (y). Tis strty urnts tt ll niors o y, wi r lry olor, v pirwis irnt olors. To s tis, lt y 1 < L y 2 ny two niors o y, wry 1,y 2 < L y. Tn y 1 R L,2 (y 2 ), so y 2 s n olor irntly rom y 1. It ollows rom Osrvtion 1 tt t otin olorin o G is nrt n n (G) ol 2 (G). In tis ppr w sll us two-olorins to t n iint oun on t nrt

3 THE TWO-COLORING NUMBER AND DEGENERATE COLORINGS 3 romti numr. It is prov in [9] tt plnr rps v two-olorin numr t most 10. Currntly, our min rsult ivs t st uppr oun or t two-olorin numr o vry plnr rp G, n nort oun on (G). Torm 1. I G is plnr rp, tn ol 2 (G) 9, n tror lso (G) 9. Torm 1 will prov in stion 2. In t, w will prov slitly stronr sttmnt tt tr is linr orrin L o V (G) su tt or vrtx x, R L,1 (x) 5n R L,2 (x) 8. S Torm 2. An xmpl ivn in [9] sows tt tr r plnr rps wos 2-olorin numr is 8. Tt xmpl is quit omplit. Hr w iv mu simplr on. Consir iv-onnt trinultion T o t pln in wi no two vrtis o r 5 r jnt. It is wll known n sy to s tt tr r ininitly mny su trinultions. Lt L linr orrin o vrtis o T,nltx t lst vrtx tt s lrr nior wit rspt to L. Suppos tt y 1,...,y k r t niors o x su tt x< L y i or i =1,...,k,nltN (x) tsto niors o x istint rom y 1,...,y k. By t oi o x, ll niors o y 1,...,y k r 2-rl rom x. Not tt y i stmosttwoniorsinn (x). I k = 1, it is sy to s tt N (x) n t niors o y 1 ontin t lst svn vrtis istint rom x. Tus, R L,2 (x) 7. On t otr n, i k 2, tn y 1 n y 2 v only x n possily on nior o x s ommon niors. I on o tm s r t lst 6, tn ty v t lst svn niors istint rom x, n n R L,2 (x) 7. I ty ot v r 5, tn x s r t lst 6, n it is in sy to s tt R L,2 (x) 7. Tis sows tt ol 2 (T ) 8. T ollowin rmins llnin opn prolm. Qustion 1. Is it tru tt vry plnr rp G stisis ol 2 (G) 8? u y F 1 F 2 x v Fi. 1. x n y r opposit wit rspt to = uv. T sontool win us to ontrolnrywn lin wit rps m in surs is s on t notion o ionl olorins in low. Lt G pln rp, n lt = uv n o G. Suppos tt t s F 1 n F 2 inint wit r ot trinls. I x, y r t vrtis istint rom u, v on t ounry o F 1 n F 2, rsptivly, tn w sy tt x, y r opposit wit rspt to (s Fiur 1). Vrtis x n y o G r si to opposit i ty r opposit wit rspt to som o G. Not tt or vrtx o r k, tr r t most k vrtis tt r opposit to it. Bout t l. [5] introu t notion o ionl olorins o pln rps, or wi on rquirs tt ny two jnt or opposit vrtis riv irnt olors. Ty propos t ollowin onjtur. Conjtur 2 (Bout t l. [5]). Evry pln rp s ionl 9-olorin.

4 4 KIERSTEAD, MOHAR, ŠPACAPAN, YANG, AND ZHU Fi. 2. A trinultion wi ns nin olors. T rp sown in Fiur 2 is n xmpl wi ns nin olors. Bout t l. prov in [5] tt 12 olors sui in nrl. Boroin [4] sow ow to sv on olor, n Snrs n Zo [12] prov tt 10 olors sui. S lso [8, Prolm 2.15]. 2. T two-olorin numr. First w n som proprtis o pln trinultions. Suppos G =(V,E) is trinultion wos vrtis r prtition into two susts U n C, wrc is n inpnnt st n vrtx in C s r 4. For vrtx x, lt U (x) t numr o niors o x in U, nlt C (x) t numr o niors o x in C. Osrvtt C (x) =4 C. x U Lt w(x) = U (x)+ C (x)/2 twit o x. Tn (1) 2 E = G (x) = G (x)+4 C = w(x)+6 C. x V x U x U Eulr s ormul implis tt E < 3 V =3 U +3 C. Tis implis tt 6 U > x U w(x), so tr is vrtx x U wit w(x) < 6 (n w(x) 5.5). ForrpG n x, y V (G) wwritx y wn x is jnt to y in G n x y otrwis. W sy tt x U is vrtx o typ (, ) i U (x) = n C (x) =. Ix y, x is o typ (5,0) or (5,1), n y is o typ (5,0), (5,1), or (5,2), tn (x, y) is ll oo pir. I xyz is trinl in G, z is o typ (5,0) or (5,1), n x, y r ot o typ (6,0), tn (x, y, z) isoo tripl. Ix, y r o typs (5,0) or (5,1) n z is o typ (6,1) n x z,y z,x y, tn(x, y, z) is lso ll oo tripl. Lmm 1. Lt G =(V,E) plnr trinultion. Suppos tt C V is n inpnnt st wr vrtx o C s r 4, ltu = V C, n suppos tt U (x) 5 or ll x U. TnG s oo pir or oo tripl. Proo. Assum tt tr is no oo pir n no oo tripl. Osrv tt U. For x U, lt(x) =w(x) 6tinitil r o x. A vrtx x is ll mjor vrtx i (x) > 0, n vrtx x wit (x) < 0 is ll minor vrtx. Eulr s ormul implis tt E =3 V 6, n totr wit (1) w onlu tt (2) (x) = w(x) 6 U =2 E 6 C 6 U = 12. x U x U

5 THE TWO-COLORING NUMBER AND DEGENERATE COLORINGS 5 I x is minor vrtx o typ (5,0), tn w lt mjor nior o x sn r o 1/3 tox. I x is minor vrtx o typ (5,1), tn mjor nior o x is sk to sn r o 1/6 tox. Lt us onsir t rsultin r (x) or ll vrtis o G. W sll sow tt (x) 0orx, wi will ontrit (2) n omplt t proo. Not tt vry vrtx, wi is not mjor vrtx, is o typ (5, 0), (5, 1), (5, 2), or (6, 0). I x is minor vrtx, tn its typ is (5, 0) or (5, 1). Sin tr r no oo pirs, x is not jnt to vrtx o typ (5, 0), (5, 1), or (5, 2). Aitionlly, sin tr r no oo tripls, x is not jnt to two jnt vrtis o typ (6, 0). It ollows tt vry minor vrtx s t lst tr mjor niors. Tror (x) 0 or ll minor vrtis x. Suppos now tt x is mjor vrtx. Sin tr r no oo pirs, no two minor vrtis r jnt, n n x sns r only to nonniorin vrtis. I minoru-nior y o x is jnt to C-nior o x, tny is o typ (5,1), n n x sns to y only 1/6 r. A U-nior jnt to two C-niors o x rivs 0. Tror t totl r snt out rom x is t most U (x)/2 /3. Hn (x) 0i U (x) 7. Assum now tt U (x) =6. I C (x) 2, tn it is sy to s tt t r snt out rom x is t most 2/3, n n (x) 0. So, ssum U (x) =6n C (x) =1. Ix s t lst two minor niors, tn ts two minor niors r not jnt (or otrwis w v oo pir). But tn w v oo tripl, ontrry to our ssumption. Tus x s t most on minor nior, n n t r snt out rom x is t most 1/3. Sin (x) =1/2 w onlu tt (x) 1/2 1/3 > 0. I (x) = 0, tn no r is snt out rom x, n n (x) =0. Nowvry vrtx s nonntiv r, wi is ontrition. Lt G =(V,E) rp n C V.AC-orrin o G is prtil orrin L o V (G) su tt t ollowin onitions ol: (i) T rstrition o L to C is linr orrin o C. (ii) T vrtis in V C r inomprl (tt is, nitr x< L y nor y< L x or x, y V C). (iii) y< L x or vry x C n y V C. Lt L C-orrin o G n x C. TstR L,2 (x) iststolly< L x, su tt itr y x or tr xists z C wit x z,y z, nx< L z.int lttr s w sy tt y is two-rl rom x wit rspt to L. Suppos tt v is vrtx o G n C = C {v}. IL is C -orrin o G n L is C-orrin o G su tt L n L r qul on C n v< L u or ll u C, tn lrly R L,2 (x) = R L,2(x) or ll x C. Tus, xtnin t C-orrin to C -orrin os not inrs t siz o ny st R L,2 (x). Torm 2. I G is plnr rp, tn tr is linr orrin L o V (G) su tt or vrtx x, R L,1 (x) 5 n R L,2 (x) 8. Inprtiulr,ol 2 (G) 9. Proo. W n to in linr orrin L o V (G) su tt or vrtx x, R L,1 (x) 5n R L,2 (x) 8. Lt C V n U = V C. W sy tt C-orrin L o G is vli i vrtx x C s t most our niors in U n R L,2 (x) 8. In t ollowin w sll prou squn o vli orrins o G so tt t orr st C oms lrr n lrr, n vntully prous linr orrin o V (G). At t innin C = (wi is rtinly vli orrin). Suppos w v

6 6 KIERSTEAD, MOHAR, ŠPACAPAN, YANG, AND ZHU vli C-orrin L o G n U. To prou lrr vli orrin w n t rp G s ollows. First w lt ll s twn vrtis o C. Ix C s t most tr niors in U, tn lt x n s twn pir o t U-niors o x. Osrv tt tis oprtion prsrvs plnrity o t rp. I x C s our niors in U, tn s in t yli orr (orin to t pln min) so tt t our niors orm 4-yl. T rsultin rp G 1 is plnr n vry vrtx o G 1 in C is o r 4 n is ontin only in trinulr s. Tror w n s only mon vrtis o U to otin pln trinultion G in wi t vrtx st F = C V (G ) is inpnnt n ll its vrtis v r 4. Sin G is trinultion n F is n inpnnt st o G, w know tt or vrtx x U, U (x) F (x). I tr is vrtx x U wit U (x) 4, tn xtn t orrin L y lttin x t nxt orr vrtx. T otin orrin L is n orrin wit x< L y or ll y C n u< L x or ll u U {x}. Lt z nior o x, wi is in C. Iz F, tn it s our niors in U n t most on o tm is not x or nior o x. Consquntly t most on nw vrtx is two-rl rom x trou z. On t otr n i z C F,tnz s t most tr niors in U n ll o tm r itr x or its niors in G. W inr tt R L,2 (x) 8 ols in t oriinl rp G. By ssumption, x s t most our niors in U, so t xtn orrin is vli. j x y i r q p x k z o y j i n m n k s m l l Fi. 3. A s o n xtnsion o orrin, wr ot x n y r o typ (5,1). Dionl vrtis r init y rokn lins. Vrtis o F r ull; vrtis o U r ollow. Suppos now tt U (x) 5inG or x U. W will pply Lmm 1 or G n t st F plyin t rol o C. T lmm implis tt tr is itr oo pir (x, y) or oo tripl (x, y, z). I tr is oo pir (x, y), tn w xtn t orrin y rqustin tt or ny u U {x, y} n v C w v u< L x< L y< L v. W lim tt t otin orrin L is vli in G or t prorr st C 1 = C {x, y}. W trt t s wn x n y r o typ (5,1) in til, n lv ll otr ss to t rr. Lt t nior o x in F,nlt t nior o y in F (possily = ). Not tt n ror4ing n tt o tm s t most on nior tt is two-rl rom x (rsp., y) n is not nior o x (rsp., y). An xmpl o tis s is sown in Fiur 3. Not tt t typ (5,1) n t rwin in Fiur 3 r wit rspt to G. T uniqu niors o x n y in F r osn in Fiur 3 s vrtis n, rsptivly, ut ty n ny otr niors, inluin possily in on o t ommon niors o x n y. Sin t xtnsion

7 THE TWO-COLORING NUMBER AND DEGENERATE COLORINGS 7 L o L is in so tt x< L y, w s tt t numr o niors u o x wit u< L x is our (in Fiur 3 ts r,,, n). Morovr, t numr o vrtis tt r two-rl in G rom x (n r irnt rom t U-niors o x in G ) is t most tr, on trou n two mor trou y (ts r j,, n). On t otr n, y s iv niors u wit u< L y (nmly, x,,, n) n on itionl two-rl vrtx trou tt is not its nior in G,ttisi. W prov tt R L,2(x), R L,2(y) 7nox, y s t most our niors in U {x, y}. Tror t xtn orrin is vli. Otr oniurtions, wn (x, y) is oo pir, r trt similrly. I (x, y, z) is oo tripl su tt x n y r o typs (5,0) or (5,1) n z is o typ (6,1),tnxtntC-orrin L to L y inin positions or x, y, nz s ollows: or ny u U {x, y, z} n v C, wstu< L x< L z< L y< L v. I (x, y, z) is oo tripl su tt x n y r o typ (6,0) n z is o typ (5,0) or (5,1),tnxtntC-orrin L to L y sttin u< L x< L y< L z< L v or u U {x, y, z} n v C. As or, it is sy to vriy tt R L,2(x), R L,2(y), R L,2(z) 8nox, y, z s t most our niors in U {x, y, z}. W onlu tt L is vli orrin. Not lso tt t nwly in prtil orr L stisis R L,1(x) 5 or ll x. On o t ss is sown in Fiur 3 or t onvnin o t rr. A slit n in t ov proo yils n pplition to nrt ionl list olorins usin t most 12 olors t vrtx. Corollry 1. Evry pln rp s nrt ionl list olorin rom lists o siz t lst 12. Proo. Lt L C-orrin, n lt RL,2 (y) t st o ll vrtis tt r two-rl rom y, totr wit ll vrtis x, su tt x< L y n x is opposit y. For t purpos o tis proo w sy tt L is vli orrin i or vrtx x C, RL,2 (x) 11, n x s t most our niors in U. Wprovtxistn o linr orr L so tt RL,2 (y) 11 or vrtx y V (G). W pro nloously s in t proo o Torm 1, tt is, w onstrut squn o vli orrins until w vntully t linr orrin o V (G). So suppos tt w r ivn vli C-orrin L, wrc V (G). Tn n t rp G s in t proo o Torm 1 to otin pln rp G. It is sy to s tt t onstrution o G ws on so tt i x, y U n x is opposit y in G, tnx is itr jnt to or opposit y in G. Ain w rriv t w ss, wr w v to trmin ow to xtn t orrin L. I tr is vrtx x wit U (x) 4inG, tn xtn t orrin L to L so tt t nxt orr vrtx is x n osrv tt ny vrtx tt is two-rl rom x n is not nior o x is lso opposit to x. It ollows tt RL,2(y) 8 ols in G. Otrwis G s itr oo pir or oo tripl. I (x, y) is oo pir, tn lt x< L y. I x, y r o typ (5,1) (s Fiur 3), tn osrv tt x s our jnt vrtis smllr tn x (ts r,,, ), two vrtis wi r tworl n ionl t t sm tim (ts r j n ), two ionl vrtis wi r not two-rl (ts r m n n) n on otr two-rl vrtx (). Tis is nin ltotr, n n RL,2(x) 9. It is sy to k tt RL,2(y) ={, x,,,, i, k, l, }, so its siz is 9. Otr ss o oo pirs r trt similrly ut RL,2 (x) n R L,2(y) r lwys t most 10. I (x, y, z) is oo tripl su tt x n y r o typs (5,0) or (5,1) n z is o typ (6, 1), w xtn L to L so tt z < L x< L y (s Fiur 3). Tn

8 8 KIERSTEAD, MOHAR, ŠPACAPAN, YANG, AND ZHU w v RL,2 (z) ={, k,,, l, s,, j,, }. Similrly R L,2(x) ={k, z,,,,, q, r, p} n RL,2(y) ={,, z, k, j, m, n, k, o}, n n in ny s t siz is lss tn 10. I (x, y, z) is oo tripl wr z is o typ (5,0) or (5,1) n x, y r o typ (6,0), tn lt x < L y < L z, n it is sy to k tt RL,2 (x), R L,2 (y), RL,2(z) Dnrt ionl 11-olorins. T ollowin torm is n improvmnt o Corollry 1 y on olor or t nonlist vrsion o nrt n ionl olorins. W v i to inlu it spit t t tt it is slitly lonr sin t mtos us in t proo o tis rsult r irnt rom t mtos o t prvious stion. Torm 3. Evry pln rp s nrt ionl olorin wit 11 olors. Proo. Suppos tt t torm is not tru. Lt G minimum ountrxmpl (wit rspt to t numr o vrtis). W my ssum tt G is trinultion o t pln. Clim 0. G s no vrtis o r 3. Proo. Suppos, on t ontrry, tt v V (G) is vrtx o r 3. Sin G is minimum ountrxmpl, tr is nrt olorin o G v, su tt no two opposit vrtis r olor wit t sm olor. A sir olorin o G is otin rom t olorin o G v, y olorin t vrtx v wit olor irnt rom t olors o its niors n its opposit vrtis. Not tt tis is lwys possil, sin tr r t most six niors or opposit vrtis o v; tror in t st o 11 olors tr r 5 possil olors or v. Tis is ontrition to t oi o G. Clim 1. G s no vrtis o r 4. Proo. Suppos, on t ontrry, tt v is vrtx o r 4. S Fiur 4. Witout loss o nrlity ssum tt is not n in G. Dlt t vrtx v, t, n ll t otin rp G. By t minimlity o G, tris nrt olorin o G su tt n r olor wit istint olors (sin ty r opposit). Tis olorin inus nrt olorin o G v. InG tr r t most our vrtis opposit to v n tr r our vrtis jnt to v. W olor t vrtx v wit olor irnt rom olors o t vrtis jnt or opposit to v. Sin,,, n r olor y pirwis istint olors, w inr rom Osrvtion 1 tt t otin olorin is nrt. Morovr, ny two opposit vrtis r olor wit istint olors. v Fi. 4. A vrtx o r 4. Lt us now introu som nottion. I G s sprtin 3-yl, tn lt C sprtin 3-yl wit t lst numr o vrtis in its intrior. Otrwis lt C t 3-yl on t ounry o t ininit. Dnot y C t rp inu y vrtis o C n tos in t intrior o C. I C ontins 4-yl or 5-yl wit t lst two vrtis in its intrior, tn lt D t on wit t lst numr o vrtis in its intrior; otrwis, lt D = C. Lt D t rp inu y t

9 THE TWO-COLORING NUMBER AND DEGENERATE COLORINGS 9 vrtis o D n tos in t intrior o D. W sll not y int(d) tsto intrnl vrtis o D. Clim 2. In C, no two intrnl vrtis o r 5 r jnt. Proo. Suppos, on t ontrry, tt u, v r jnt intrnl vrtis o r 5; s Fiur 5 or nottion. Sin C s no sprtin 3-yls,,, n r not s o G. Dlt t vrtis u n v, t s,, n, nllt otin rp G (s Fiur 5). u v Fi. 5. Two intrnl jnt vrtis o r 5. Osrv tt,, n, r two pirs o opposit vrtis in G ; tror tr xists nrt olorin o G su tt t vrtx sts {,,, } n {,,, } r olor y pirwis istint olors. Sin ot u n v r vrtis o r 5, tr r t most nin (iv on opposit n our on jnt vrtis) olors proiit or u n v. Tror, olorin o G n otin rom t olorin o G y olorin u n v y istint vill olors. Osrv tt t otin olorin is nrt (y Osrvtion 1) n tt ny two opposit vrtis r olor y istint olors. v u z x, i i Fi. 6. u, v, z, nx r vrtis o rs 5, 6, 6, n6, rsptivly. Clim 3. An intrnl vrtx u o D o r 5 nnot jnt to tr intrnl vrtis v, x, z o D o r 6, wrz is jnt to x n v. Proo. Suppos tt u, v, z, nx r intrnl vrtis o D ontritin t lim (s Fiur 6). W lim tt vrtis i n r nitr jnt nor v ommon nior. Assum (or ontrition) tt ty r jnt n tt t i is m s sown in Fiur 6. Consir t 4-yl E = iuv n osrv tt it ontins t lst two vrtis in t intrior. By t minimlity o D, t 4-yl E is not ontin in D. So ssum tt t is vrtx o D ontin in t intrior o E. I t is jnt to ot n i, tntvuit is 5-yl ontritin t minimlity o D. Otrwis, tr r two vrtis o D ontin in t intrior o E n vrtx t V (D) in t xtrior o E; intisst is jnt to n

10 10 KIERSTEAD, MOHAR, ŠPACAPAN, YANG, AND ZHU i. Ain w onlu tt t vuit is 5-yl, ontritin t minimlity o D. Anloous rumnts prov tt n i nnot v ommon nior. Not lso tt t vrtis,,...,i sown in Fiur 6 r pirwis istint, sin otrwis D woul ontin sprtin 3-yl or 4-yl (irnt rom D). Morovr, t sm rumnts sow tt non o,,,, n is n o G. Lt us lt vrtis u, v, z, nx n intiy i n. Furtr, t s,, n s sown in Fiur 6, n ll t otin rp G. As prov ov, G is rp witout loops or multipl s. By t minimlity o G, tr is nrt olorin o G, su tt o t vrtx sts {i,,, }, {,, }, {,,, }, n{, i} is olor y pirwis istint olors (not tt i,, n, r pirs o opposit vrtis in G ). Tis olorin inus olorin o ( surp o) G, wri n riv t sm olor. Sin n i r t istn t lst 3inG, tis olorin is nrt n s opposit vrtis olor wit istint olors. W now olor t vrtis x, z, v, nu in tis orr. Sin x is olor or u, v, nz n t olor o quls t olor o i, w in tt tr r t most nin proiit olors or x; ts r t olors us or t vrtis,,,,, i, n tr itionl opposit vrtis istint rom, v, n. W olor t vrtx x wit on o t two rminin olors not proiit or x. Nxt w olor z, wi s 10 proiit olors; ts r t olors us or t vrtis,,, x,,,,, nt two opposit vrtis wit rspt to s n. Notttwvrqust tt t olor o z irnt rom in orr to l to pply Osrvtion 1 wn olorin v. So tr is olor not proiit or z, n w us tis olor to olor it. Sin i n v t sm olor, t olor o x is istint rom t olor o. Hn, t olors on t niors o z r ll istint. Tror t olorin is still nrt. Nxt w olor v, wi s t most 10 proiit olors: it s six niors n six opposit vrtis, ut u is not yt olor, n, i v t sm olor. Finlly, w olor u, wi s lso t most nin proiit olors (t olors o iv opposit n iv jnt vrtis, wr two o tm oini, nmly t olors o n i). By pplyin Osrvtion 1 t stp, w onlu tt t rsultin olorin o G is nrt. It ws onstrut in su wy tt ll opposit vrtis v irnt olors. Tis ontrition to nonolorility o G omplts t proo o Clim 3. Clim 4. An intrnl vrtx u o D o r 7 nnot jnt to tr intrnl vrtis v, x, z o D o r 5 n two vrtis y n t o r 6, su tt y is jnt to v n x n t is jnt to x n z. Proo. Suppos tt u, v, x, z, y, t r vrtis o D ontritin t lim. S Fiur 7 or itionl nottion. Sin u, v, z, nx r intrnl vrtis o D, w in tt n r nitr jnt nor v ommon nior (sin otrwis on woul in 4- or 5-yl wit wr vrtis in t intrior wn ompr to D). Lt us rmov vrtis t, u, v, x, y, z. Tn intiy vrtis n, s i, i,, n (or ), n ll t otin rp G. T nrt olorin o G inus nrt olorin o surp o G, wr n riv t sm olor. W will olor t vrtis u, y, t, x, v, z in tis orr. Osrv tt or t vrtx u, tr r t most 10 = proiit olors: 14 olors or jnt n opposit vrtis, +2 olors or j n (wi w wnt to olor irnt rom u to us Osrvtion 1 wn ltr olorin y n t), 5 or not yt olor jnt vrtis, n 1 us t olor o is qul to t olor o. Similrly on n

11 THE TWO-COLORING NUMBER AND DEGENERATE COLORINGS 11 i v y u t z i j x j Fi. 7. u, v, z, nx r vrtis o rs 7, 5, 5, n5, rsptivly. ru tt or y tr r t most it rstritions, n or t, v, x, z, tr r t most 10 proiit olors. Sin w olor wit 11 olors n or vrtx u, y, t, x, v, z tr r wr tn 11 proiit olors, w n xtn t prtil olorin o G inu y t olorin o G to nrt ionl olorin o G. Clim 4 implis tt, i u is vrtx o r 7 wit tr (nonjnt) niors v, x, z o r 5 n istriut s sown in Fiur 7, tn itr y or t s r t lst 7. W omplt t proo o t torm y pplyin t isrin mto in D. W in t r (v) ovrtxv V ( D) s { D(v) 6 i v int(d), (3) (v) = D(v) 3 i v V (D). It ollows rom t Eulr ormul tt ( D(v) 6) = 6 2 D, n tus (4) v V ( D) (v) = v V (D) v V ( D) ( D(v) 3) + v int(d) ( D(v) 6) = D 6. W now prorm t isrin prour s ollows: (i) Evry vrtx o D ivs 1 to nior o r 5 in int(d). (ii) Evry vrtx in int(d) o r t lst 8 ivs 1/2 to nior o r 5inint(D). (iii) Evry vrtx in int(d) o r 7 wit t most two niors o r 5 in int(d) ivs 1/2 to nior o r 5 in int(d). (iv) A vrtx u int(d) o r 7 wit tr niors o r 5 in int(d) s its nioroo s sown in Fiur 7, xpt tt y n t r not nssrily o r 6. () I (y) 7 n (t) 7, tn u ivs 1/2 to x n 1/4 to v n z. () I (y) 7 n (t) =6,tnu ivs 1/2 to z n 1/4 to v n x. () I (y) = 6 n (t) 7, tn u ivs 1/2 to v n 1/4 to z n x. Not tt vrtx o D nnot v r 3in D, sin tis woul imply tt D is not 3-, 4-, or 5-yl wit minimum numr o vrtis in its intrior.

12 12 KIERSTEAD, MOHAR, ŠPACAPAN, YANG, AND ZHU Morovr, i v is vrtx o D n s α>1 niors o r 5 in int(d), tn t r o v in D is t lst 2α + 1 y Clim 2. Tis implis tt vry vrtx o D rtins (tr isrin) t lst 0 o its r. Also, y Clim 2, vrtx o r α 7 s t most α 2 niors o r 5. So w inr rom t isrin ruls tt vry vrtx o int(d) or 7 s nonntiv r tr t isrin prour. Sin intrnl vrtis o r 6 kp tir r t zro, n tr r no vrtis o r lss tn 5; t only nits or vin ntiv r r intrnl vrtis o r 5. Osrv tt vry vrtx v int(d) o r 5 is itr jnt to vrtx o D or s, y Clims 0, 1, 2, n 3, t lst two niors o r 7. I v s our or iv niors o r 7, tn, tr isrin, its r will t lst 0, sin vry vrtx o r 7 ivs t lst 1/4 to o its niors o r 5. I v s tr niors o r 7 n on o tm is not o r 7 or os not v tr niors o r 5, tn tis nior will iv 1/2 to v n tus v will v r o t lst 0. Otrwis v s tr niors o r 7, n o tm s tr niors o r 5. It ollows rom rul (iv) tt on o t niors o v will iv 1/2 to v, so t inl r o v will nonntiv. In t rminin s, wr v s xtly two niors o r 7, w s tt ts two niors r not jnt y Clim 3. Tror t isrin rul (iv) implis tt ot o tm iv 1/2 to v. Tus, t inl r t v is nonntiv. Tis isrin pross provs tt t lt-n si o (4) is nonntiv, wil t rit-n si is ntiv, ontrition. Tis omplts t proo o Torm 3. Finlly w turn to t proo o t ionl list olorin rsult. Osrv tt intiition o vrtis nnot on wit list olorins, so t proos o Clims 3 n 4 rom t ov proo nnot xtn to list olorins. W nxt iv irnt proo o Corollry 1. T proo is similr to t proo o Torm 3. W strt y ssumin t ontrry, n lt G minimum ountrxmpl. Din C, C,D,n D t sm wy s in t proo o Torm 3. T proos o Clims 0, 1, n 2 rom t proo o Torm 3 lso ol or list olorins. Clim 5. An intrnl vrtx o D o r 5 nnot jnt to two jnt intrnl vrtis o D o r 6. Proo. Suppos, on t ontrry, tt u, v, z r vrtis o rs 5, 6, n 6, rsptivly. S Fiur 8 or urtr nottion. W o t rution s ollows. v u z Fi. 8. u, v, nz r vrtis o rs 5, 6, n6, rsptivly. Dlt t vrtis u, v, nz n s,,, n. Not tt ts r not s o G, sin D s no sprtin 3-yls. Lt us ll t otin rp

13 THE TWO-COLORING NUMBER AND DEGENERATE COLORINGS 13 G. T lists or vrtis o G r inrit G. By t minimlity o G, tris nrt list olorin o G, su tt,,, n,,,, rsptivly, r olor y pirwis istint olors (sin,, n, r pirs o opposit vrtis in G ). Tis olorin inus olorin o surp o G. W nxt olor v, z, nu (in tis orr). Osrv tt v n z v t most 11 proiit olors (6 opposit vrtis n 4, rsptivly 5, niors, n t olor o is proiit or v); t vrtx u is o r 5, n tus it s t most 10 proiit olors. Tus, or ll tr rminin vrtis tr is r olor so tt w n xtn t list olorin o G to nrt list olorin o G; tis is ontrition provin t lim. Finlly, t ollowin isrin prour ls to ontrition. (i) Evry vrtx o D ivs 1 to nior o r 5 in int(d). (ii) Evry vrtx o r t lst 7 o int(d) ivs 1/3 to nior o r 5inint(D). Sin vry vrtx o r 5 o int(d) s (y Clims 0, 1, 2, n 5) t lst tr niors o r 7 or s nior in D, n vry vrtx o r α 7o int(d) s t most α 2 niors o r 5 in int(d), w onlu tt t lt-n si o (4) is nonntiv, wil t rit-n si is ntiv, ontrition. REFERENCES [1] O. V. Boroin, On omposition o rps into nrt surps, Diskrtny Anliz., 28 (1976), pp (in Russin). [2] O. V. Boroin, A proo o Grünum s onjtur on t yli 5-olorility o plnr rps, Sovit Mt. Dokl., 17 (1976), pp [3] O. V. Boroin, On yli olorins o plnr rps, Disrt Mt., 25 (1979), pp [4] O. V. Boroin, Dionl 11-olorin o pln trinultions, J. Grp Tory, 14 (1990), pp [5] A. Bout, J.-L. Fouqut, J.-L. Jolivt, n M. Riviér, On spil olourin o ui rps, Ars Comin., 24 (1987), pp [6] G. Cn n R. H. Slp, Grps wit linrly oun Rmsy numrs, J. Comin. Tory Sr. B, 57 (1993), pp [7] B. Grünum, Ayli olorins o plnr rps, Isrl J. Mt., 14 (1973), pp [8] T. R. Jnsn n B. Tot, Grp Colorin Prolms, Jon Wily & Sons, Nw York, [9] H. A. Kirst n W. T. Trottr, Plnr rp olorin wit n unooprtiv prtnr, J. Grp Tory, 18 (1994), pp [10] H. A. Kirst n D. Yn, Orrins on rps n m olorin numr, Orr, 20 (2003), pp [11] D. Rutn, A onjtur o Boroin n olorin o Grünum, Fit Crow Conrn on Grp Tory, Ustron 06, Eltron. Nots Disrt Mt., 24 (2006), pp [12] D. P. Snrs n Y. Zo, On ionlly 10-olorin pln trinultions, J. Grp Tory, 20 (1995), pp [13] C. Tomssn, Domposin plnr rp into nrt rps, J. Comin. Tory Sr. B, 65 (1995), pp [14] C. Tomssn, Domposin plnr rp into n inpnnt st n 3-nrt rp, J. Comin. Tory Sr. B, 83 (2001), pp [15] X. Zu, T m olorin numr o plnr rps, J. Comin. Tory Sr. B, 75 (1999), pp [16] X. Zu, Rin tivtion strty or t mrkin m, J. Comin. Tory Sr. B, 98 (2008), pp [17] X. Zu, Colourin rps wit oun nrliz olourin numr, Disrt Mt., to ppr.