Solvability of Cubic Graphs - From Four Color Theorem to NP-Complete

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1 Solvility of Cui Grphs - From Four Color Theorem to NP-Complete Tony T. Lee nd Qingqi Shi Stte Key Lortory of Advned Communition Systems nd Networks, Deprtment of Eletroni Engineering, Shnghi Jio Tong University, Shnghi, Chin ttlee@ie.uhk.edu.hk Deprtment of Informtion Engineering, The Chinese University of Hong Kong, Hong Kong, Chin qqshi@ie.uhk.edu.hk rxiv: [s.dm] 8 Jn 2014 Astrt Similr to Euliden geometry, grph theory is siene tht studies figures tht onsist of points nd lines. The ore of Euliden geometry is the prllel postulte, whih provides the sis of the geometri invrint tht the sum of the ngles in every tringle equls π nd Crmer s rule for solving simultneous liner equtions. Sine the ounterprt of prllel postulte in grph theory is not known, whih ould e the reson tht two similr prolems in grph theory, nmely the four olor theorem ( topologil invrint) nd the solvility of NP-omplete prolems (disrete simultneous equtions), remin open to dte. In this pper, sed on the omplex oloring of ui grphs, we propose the reduiility postulte of the Petersen onfigurtion to fill this gp. Compring edge oloring with system of liner equtions, we found tht the postulte of reduiility in grph theory nd the prllel postulte in Euliden geometry shre some ommon hrteristis of the plne. First, they oth provide solvility onditions on two equtions in the plne. Seond, the two si invrints of the plne, nmely the hromti index of ridgeless ui plne grphs nd the sum of the ngles in every tringle, n e respetively dedued from them in strightforwrd mnner. This reduiility postultion hs een verified y more thn one hundred thousnd instnes of Peterson onfigurtions generted y omputer. Despite tht, we still don t hve logil proof of this ssertion. Similr to tht of the prllel postulte, we tend to think tht desriing these nturl lws y even more elementry properties of the plne is inoneivle. Keywords edge oloring; olor exhnge; Kempe wlk; Petersen grph I. INTRODUCTION AND OVERVIEW Similr to Euliden geometry, grph theory is siene tht studies figures tht onsist of points nd lines. Insted of mesuring ngles nd distnes, grph theory fouses on the topologil onfigurtions tht re omposed of verties nd edges. The ore of Euliden geometry is the fifth postulte, ommonly lled the prllel postulte. To the nients, however, the prllel postulte ws less ovious thn the other four postultes. For lst two thousnd yers, mny tried in vin to prove the prllel postulte using Eulid s other four postultes [15]. Some flse proofs of the prllel postulte were epted for mny yers efore they were exposed. It is now known tht the prllel postulte is nturl lw of two dimensionl Euliden plnes, nd proof is impossile. This lw hs produed the following monumentl rmifitions: 1) The geometri invrint tht the sum of the ngles in every tringle equls π is diret onsequene of the prllel postulte. A generliztion of this result is the Guss-Bonnet theorem in differentil geometry. 2) The prllel postulte provides the solvility ondition of two liner equtions in the plne. The theory of determinnt nd Crmer s rule for solving simultneous liner equtions re generliztion of this ondition in Euliden spe. Sine the ounterprt of the prllel postulte in grph theory is not known, whih ould e the reson tht the theoretil proofs or solutions of two similr prolems in grph theory, nmely the four olor theorem ( topologil invrint) nd the solvility of NP-omplete prolems (disrete simultneous equtions), remin open to dte. In this pper, sed on the omplex oloring method desried in [13], we propose the reduiility postulte of the

2 Petersen onfigurtion to fill this gp. An immedite onsequene of this proposition is the 3-edge oloring theorem, or equivlently, the four olor theorem (4CT). This self-evident proposition hs een verified y more thn one hundred thousnd instnes generted y omputer. Despite tht, we still don t hve logil proof of this ssertion. The 4CT is the holy gril of grph theory, ut the proof of this simply stted theorem is elusive. The 4CT sttes tht the minimum numer of olors required to olor mp is four, whih represents topologil invrint of the plne. Ever sine the prolem ws rised y Frnis Guthrie in 1852, the theorem ws flsely proved twie y Alfred Kempe [?] in 1879, nd Peter Tit [2] in Despite their fruitless efforts, their ides provided fundmentl insights into grph oloring tht re still of prmount importne in grph theory. The 4CT ws finlly proved y Kenneth Appel nd Wolfgng Hken [3], [4] in Their proof relies on omputer-ided heking tht nnot e verified y humn. In 1997, Neil Roertson, Dniel Snders, Pul Seymour, nd Roin Thoms pulished simpler version of omputer-ssisted proof sed on the sme ide [5]. The history of 4CT is detiled in [6], [7], nd rief survey on the progress is provided in [8]. In his 1880 pper, Peter Tit proposed tht ny ridgeless ui plnr grph hs Hmiltonin yle. His proof of 4CT sed on this flse ssumption ws refuted y Julius Petersen [9] in However, it ws not until 1946 tht Willim Tutte found tht there re suh plnr grphs without ny Hmiltonin tours. The min ontriution of his pper ws to estlish the following equivlent formultion of 4CT. Theorem 1 (Tit). A ridgeless ui plnr grph G is 4-fe-olorle if nd only if G is 3-edge-olorle. A proof of this theorem n e found in mny ooks nd ppers [10], [11]. It essentilly trnsforms the 4CT from vertex-oloring prolem into n edge-oloring prolem. In omputer siene, stisfiility, revited s SAT, is the prolem of determining if there exists truth ssignment of vriles tht stisfies given Boolen formul. SAT ws the first known exmple of n NP-omplete prolem. The stisfiility prolem of Boolen expression ϕ n lso e onsidered s the solvility of set of simultneous Boolen equtions [22]. For exmple, truth ssignment of the expression ϕ = (x 1 x 2 x 3 ) (x 2 x 3 ) (x 1 x 3 ) is solution of the following set of Boolen equtions: x 1 x 2 x 3 = 1, x 2 x 3 = 1, x 1 x 3 = 1. Mny NP-omplete prolems n lso e onsidered s solving set of simultneous disrete equtions. One of them is to deide the hromti index of ui grph. The hromti index χ e (G) of simple grph G is the minimum numer of olors required to olor the edges of the grph suh tht no djent edges hve the sme olor. A theorem proved y Vizing [?] sttes tht the hromti index is either or + 1, where is the mximum vertex degree of grph G. Grph G is sid to e Clss 1 if χ e (G) = ; otherwise, it is Clss 2. Aording to Tit s equivlent formultion, the 4CT is estlished if every ridgeless ui plnr grph G is Clss 1. In [21], Holyer proved tht Boolen expression ϕ n e onverted into ui grph G, suh tht ϕ is stisfile if nd only if G is 3-edge olorle. Therefore, it is NP-omplete to determine the hromti index of n ritrry ui grph. The onept of vrile edges is introdued in the omplex oloring method; proper oloring is hieved y eliminting ll vriles in olor onfigurtion of ui grph. A onneted ridgeless ui grph tht does not hve 3-edge oloring is lled snrk. Mny importnt nd diffiult prolems in grph theory re relted to snrks. However, their properties nd strutures re still lrgely unknown. For the first time, this pper ompletely speifies the neessry nd suffiient ondition of snrks in terms of omplex oloring onfigurtions. In priniple, the entire grph G = (V, E) n e onsidered s set of simultneous equtions, in whih eh vertex v V represents onstrint on oloring of edges. The vrile elimintion proedure of edge oloring is similr to the lgeri method for solving systems of liner equtions. A omprison etween these two proedures is summrized in Tle I. The inonsisteny of two liner equtions is usully interpreted s two prllel lines in Euliden spe. In the omplex oloring of the Petersen grph, the smllest snrk, the onfigurtion ontins two vriles in two

3 TABLE I A COMPARISON BETWEEN LINEAR EQUATIONS AND EDGE COLORING System of liner equtions(euliden Geometry) Edge oloring(grph Theory) Opertions Arithmeti Opertions Color Exhnges Constrints Liner Equtions Verties nd edges Unknowns Vriles Vrile-edges Algorithms Vrile Elimintion Vrile Elimintion Solution Consisteny 3-edge oloring No solution Inonsisteny Snrk disjoint odd yles, whih n never e eliminted euse they will never meet eh other. Tht is, the two yles ehve the sme s two prllel lines in Euliden spe. The nlogy etween the prllel postulte in Euliden geometry nd the reduiility postulte in grph theory is illustrted in Fig. 1. They shre some ommon hrteristis of the plne. First, they n respetively dedue the two invrints of the plne. Seond, they oth provide solvility onditions on equtions in the plne. Prllel Postulte Angles sum of tringle equls π (Invrint of Plne) Reduiility Postulte (Petersen Configurtion) Four Color Theorem (Invrint of Plne) Theory of Determinnt Solvility of liner equtions Liner Alger Theory of Snrk (Petersen Grph) Solvility of Cui Grphs (NP-omplete) Coloring Alger () Euliden geometry. () Grph theory. Fig. 1. The two postultes nd their rmifitions. The rest of this pper is orgnized s follows. In setion II, we riefly desrie the si onept of omplex olors nd the rules of olor exhnges. In setion III, we introdue the deomposition of three-olored onfigurtion into two-olored mximl su-grphs. In setion IV, we define the reduiility of onfigurtions, from whih we show tht snrk n e hrterized y losed set of irreduile onfigurtions. In Setion V, we introdue the Petersen onfigurtion, nd propose the postulte tht Petersen onfigurtion of ridgeless ui plnr grph must e reduile. In setion VI, we show tht the 3-edge oloring theorem of ridgeless ui plnr grphs is n immedite onsequene of the reduiility postulte. The disussion in Setion VII fouses on omprison etween grph theory nd Euliden geometry, in prtiulr, the nlogy etween the reduiility postulte in grph theory nd the prllel postulte in Euliden geometry. Setion VIII provides onlusion. Furthermore, Appendix A provides exmples of ontrtion of snrks to the Petersen grph. A video lip to demonstrte the opertion of omplex oloring ws posted t YouTue t II. PRELIMINARIES An edge-oloring method sed on exhnges of omplex olors is proposed in [13]. The si ide is to prtition eh edge of grph into two links, then olor the links nd perform olor exhnges etween the links insted of the edges. This olor exhnge method is riefly desried in this setion to filitte our disussions. Even though the omplex oloring method n e pplied to ny grph, we only onentrte on the omplex oloring of ridgeless ui grphs, euse it is the fol point of this pper. A. Complex Coloring of Cui Grphs Let G = (V, E) e ui grph with vertex set V, edge set E. The inidene grph G is onstruted from G y pling fititious vertex in the middle of eh edge of G. Let E (G ) = {e i,j e i,j E(G)} denote the set of fititious verties on edges. Then edge e i,j E(G ) onsists of two links, denoted y l i,j = (v i, e i,j ) nd

4 l j,i = (v j, e i,j ), whih onnet two end verties v i nd v j of e i,j. Fig. 2() illustrtes the inidene grph of the tetrhedron. Let L(G ) e the set of links nd C = {,, } denote the set of three olors. A oloring funtion is mpping of olors on links, σ : L(G ) C. The olor of link l i,j L(G ) is denoted s σ(l i,j ) = i,j. Sine eh edge e E(G ) onsists of two links, the olor funtion n lso e onsidered s mpping defined on the set of edges, σ : E(G ) C C. Sine the oloring funtion ssigns two olors to eh edge of grph G, or one olor for eh link of the inidene grph G, the mpping σ is lled omplex oloring of grph G. The oloring funtion σ is onsistent if olors ssigned to those links inident to the sme vertex v re ll distint for ll v V (G). We define the olored edge σ(e i,j ) = e i,j = ( i,j, j,i ) = (α, β), α, β C s two-tuple olor vetor, where i,j = σ(l i,j ) = α nd j,i = σ(l j,i ) = β re respetive olors of the two links of e i,j. The vetor of olored edge e i,j = (α, β) is vrile if α β; otherwise, e i,j = (α, α) is onstnt. A proper 3-edge oloring of grph G, lled Clss 1 grph, n e hieved y eliminting ll vriles. The grph G is Clss 2 grph if vriles nnot e eliminted. As n exmple, onsistent oloring of tetrhedron ontining two (, ) vriles is shown in Fig. 2(), nd proper oloring of tetrhedron is shown in Fig. 2() with the set of olors C = {,, }, where,, nd represent red, green, nd lue olors, respetively. () Inidene grph of tetrhedron. () Consistent oloring of tetrhedron. () Proper oloring of tetrhedron. Fig. 2. Complex oloring of the tetrhedron grph. B. Kempe Wlks nd Vrile Elimintions In onsistently olored inidene grph, n (α, β) Kempe pth, or simply (α, β) pth, where α, β C nd α β, is sequene of djent links l 1, l 2,..., l n 1, l n suh tht σ(l i ) α, β for i = 1,..., n. The verties ontined in the pth re lled interior verties of the pth. There re two types of mximl (α, β) pths in ridgeless ui grph: 1) (α, β) yle: The two end-links l 1 nd l n re djent to eh other. 2) (α, β) open pth: The two end-links l 1 nd l n re not djent to eh other, nd oth ends of the pth re fititious verties. An (α, β) vrile edge is lwys ontined in mximl (α, β) pth, either n (α, β) yle or n (α, β) open pth. Vrile elimintions n e hieved y the inry olor exhnge opertion performed on two djent olored edges e j,i = ( j,i, i,j ) nd e i,k = ( i,k, k,i ), whih is defined s follows: ( j,i, i,j ) ( i,k, k,i ) = ( j,i, β) (α, k,i ) ( j,i, α) (β, k,i ). (1) If the two djent edges re vriles e j,i = (α, β) nd e i,k = (α, γ), then the following olor exhnge opertion: e j,i e i,k = (α, β) (α, γ) (α, α) (β, γ) (2) n eliminte one of these vriles. In generl, vrile elimintions require sequene of olor exhnges to move one vrile to nother vrile long two-olored Kempe pth. The Kempe wlk of (α, β) vrile on (α, β) pth is sequene of olor exhnge opertions performed on its interior verties. Exmples of vrile elimintions y Kempe wlks re provided in Fig. 3. Consider the (, )

5 pth (, ) (, ) (, ) shown in Fig. 3. The vrile e 1 = (, ) n wlk to nother vrile e 2 = (, ) y the following sequene of olor exhnges performed on its interior verties: (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ), (3) in whih two vriles re eliminted y olor exhnges. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Fig. 3. Vrile elimintions y Kempe wlk. Another useful pplition of the Kempe wlk is to negte vrile. The negtion of vrile (α, β), denoted s (α, β) = (β, α), represents olor vetor in the opposite diretion of (α, β). The negtion requires the olor onversion of the entire mximl (α, β) pth H tht ontins the vrile (α, β). Tht is, the olor inversion of the (α, β) pth H involves sequene of olor exhnges of ll interior verties of H. The negtion only pplies to mximl (α, β) pth, either (α, β) yle or n open (α, β) pth; otherwise, the opertion my introdue new vriles. III. DECOMPOSITION OF A CONFIGURATION As the previous setion shows, vriles n move long two-olored lternte pths vi olor exhnge opertions, lled Kempe wlks, nd they will nel eh other while moving round the grph. The prolem is solved if ll vriles re eliminted nd proper 3-edge olored ui grph G is rehed; otherwise, G is Clss 2 ui grph, lled snrk. Our study of edge oloring of ui grphs strts with Petersens theorem stted s follows. Theorem 2 (Petersen). Every ridgeless ui grph ontins perfet mthing. This theorem first ppered in [14]. Tody, it n e proved y n pplition of the Tutte theorem [10], [11]. In ridgeless ui grph G with perfet mthing, the edges tht re not in the perfet mthing form set of disjoint yles, lled Tit yles. For ny Petersen s perfet mthing, ssigning olor to the edges in the perfet mthing nd olor or to the links in these Tit yles, we n otin omplex oloring T (G) of grph G tht only ontins (, ) vriles. It is esy to show tht ll (, ) vriles re ontined in odd (, ) Tit yles, nd every odd (, ) Tit yle ontins extly one (, ) vrile in suh omplex oloring. The omplex oloring T (G) orresponding to the Petersen s perfet mthing is lled onfigurtion of grph G in this pper. As n exmple, two onfigurtions of ui plnr grph G re depited in Fig. 4. The onfigurtion shown in Fig. 4() hs two (, ) vriles respetively ontined in two disjoint odd (, ) yles. The properly olored onfigurtion shown in Fig. 4() ontins one even (, ) yle, one even (, ) yle, nd two even (, ) yles. For ny given Petersen s perfet mthing of ridgeless ui grph G, eh (, ) Tit yle n e olored in two different wys. One oloring of the (, ) Tit yle is the negtion of the other, mening exhnge the two, olors. And eh vrile e = (, ) n e loted t ny edge of the odd yle tht ontins e. Therefore, different olor ssignments orresponding to the sme Petersen s perfet mthing form n equivlent lss, nd they re onsidered s the sme onfigurtion T (G) ut in different, oloring sttes. Tht is, onfigurtion T (G) of grph G is uniquely determined y the set of (, ) edges, Petersen s perfet mthing, while the stte of T (G) is determined y nd links. Let τ o nd τ e e the respetive numer of odd nd even (, ) Tit yles in T (G), nd S T (G) denote the set of sttes of T (G). Then we hve: S T (G) = 2 τ n 1 n 2 n τo, (4)

6 () Two disjoint odd (, ) yles. () (, ), (, ), nd (, ) even yles. Fig. 4. Complex oloring onfigurtions of ui plnr grph G. where τ = τ o + τ e is the totl numer of (, ) Tit yles, nd n i is the length of i-th odd (, ) Tit yles, for i = 1,..., τ o, in the onfigurtion T (G). Tht is, there is one-to-one orrespondene etween perfet mthings nd onfigurtions of ridgeless ui grphs. In onfigurtion T (G) of ridgeless ui grph G, ny two mximl two-olored su-grphs H 1 (α, β) nd H 2 (α, β), for α, β C = {,, }, must e vertex disjoint, whih implies tht ny onfigurtion n e deomposed into set of mximl two-olored (, ), (, ) nd (, ) su-grphs. There re five suh kinds of su-grphs in onfigurtion, nd they re listed s follows: (, ) Tit yles, they n e either odd or even, nd eh odd Tit yle ontins n (, ) vrile. (, ) nd (, ) even yles. (, ) nd (, ) open pths onneting (, ) vriles. The olletion of these mximl two-olored su-grphs overs eh link of grph G extly twie, s illustrted y the onfigurtion T (G) of ui grph G shown in Fig. 5(). The properties of these mximl two-olored su-grphs re desried s follows: 1) Loking Cyle The odd Tit yle H(, ) tht ontins n (, ) vrile is lled loking yle. Two (, ) loking yles of the onfigurtion T (G) re shown in Fig. 5(). 2) Exlusive Chin The open (, ) nd (, ) pths onneting two (, ) vriles re lled exlusive hins. Negting n (, ) exlusive hin will hnge the two end (, ) vriles into two (, ) vriles. Similrly, olor inverting of (, ) exlusive hin will hnge the two ends into two (, ) vriles. Therefore, the olor inversion of ny exlusive hin is prohiited, nd the nme exlusive hin implies tht the two end vriles re mutully exlusive y the hin. The (, ) exlusive hin nd the (, ) exlusive hin of T (G) re shown in Fig. 5() nd 5(f), respetively. 3) Resolution Cyle A vrile ontined in loking yle H(, ) n e relesed y negting nother yle R 1 (, ) whih shres some (, ) edges with H(, ). After inverting the olors in yle R 1 (, ), the shred (, ) edges eome (, ) edges, whih deform the originl loking yle H(, ). Thus, yle R 1 (, ) is lled resolution yle of the (, ) vrile loked in the loking yle H(, ). After inverting the olors of the resolution yle R 1 (, ), if the loked (, ) vrile n e neled with nother (, ) vrile y wlk on newly reted Kempe pth, then R 1 (, ) is lled n essentil yle. Otherwise, it is nonessentil yle. Similrly, yle R 2 (, ) tht shres some (, ) edges with H(, ) lso serve s resolution yle of the (, ) vrile ontined in H(, ). The deomposition of onfigurtion T (G) of grph G is illustrted y the running exmple shown in Fig. 5, from whih we oserve the following dditionl properties of these mximl two-olored su-grphs of T (G): 1) The (, ) exlusive hin nd the (, ) resolution yle re disjoint, s shown in Fig. 5() nd 5(d), respetively, ut their union onsists of ll nd links of T (G). 2) A similr property holds for nd links of T (G), s shown in Fig. 5(f) nd 5(g).

7 () Configurtion T (G). () Two loking (, ) yles. () The (, ) exlusive hin. (d) An essentil (, ) yle. (e) Two even (, ) yles fter negting (, ) yle. (f) The (, ) exlusive hin. (g) A nonessentil (, ) yle. (h) Two odd (, ) yles fter negting (, ) yle. Fig. 5. Deomposition of omplex oloring onfigurtion T (G). 3) The olletion of ll links in the two loking (, ) yles, the (, ) exlusive hin, the (, ) exlusive hin, the (, ) resolution yle, nd the (, ) resolution yle inludes every link in T (G) extly twie. Next, we onsider the negtion of the two resolution yles, one (, ) yle nd one (, ) yle, of the onfigurtion T (G), shown in Fig. 5(d) nd 5(g), respetively. If we negte the (, ) yle, the resulting onfigurtion hs two even (, ) yles, s shown in Fig. 5(e), nd the two (, ) vriles ontined in the sme (, ) yle n e esily eliminted y Kempe wlk. Thus, the (, ) resolution yle displyed in Fig. 5(d) is n essentil yle. On the other hnd, the oloring resulting from the negtion of the (, ) yle still hs two disjoint odd (, ) yles, eh of whih ontins n (, ) vrile, s shown in Fig. 5(h). Therefore, the (, ) resolution yle displyed in Fig. 5(g) is nonessentil yle. IV. REDUCIBILITY OF A CONFIGURATION A onfigurtion T (G) of ridgeless ui grph G is uniquely determined y the (, ) edges in Petersen s perfet mthing, ut the sttes of T (G) re determined y nd links. Thus, the following two opertions will use stte trnsitions of T (G) ut not hnge the onfigurtion itself: Negte ny (, ) yle, either even or odd. Move ny (, ) vrile within its loking yle. Notie tht the stte trnsitions due to the ove two opertions will retin the su-grphs of ll (, ) yles intt; however, they will hnge (, ) nd (, ) exlusive hins nd resolution yles.

8 Let S T (G) denote the set of ll sttes of T (G). We sy tht stte ξ S T (G) is reduile if one of the even (, ) or (, ) yles in the stte ξ is essentil; otherwise, the stte ξ S T (G) is irreduile. If the stte ξ is irreduile, then we n implement the ove stte trnsition opertions to hnge the stte of T (G) to nother stte. Sine the numer of sttes is finite, given y S T (G) = 2 τ n 1 n 2... n τo, it is fesile to verify the reduiility of ll sttes of T (G) in systemti mnner y deterministi lgorithm. A onfigurtion T (G) is reduile if one of the sttes ξ S T (G) is reduile, mening tht two (, ) vriles in T (G) n e eliminted y Kempe wlk fter negting n essentil yle in the stte ξ. The resulting onfigurtion T (G) is proper oloring of grph G if it no longer ontins ny remining (, ) vriles. By ontrst, onfigurtion T (G) is irreduile if ll sttes re irreduile. An irreduile onfigurtion T (G) does not imply tht the underlying ui grph is Clss 2, euse onfigurtion T (G) n e trnsformed into nother onfigurtion T (G) y negting some even (, ) or (, ) yles in T (G). By olleting the ove disussions, s illustrted y the trnsition digrm shown in Fig. 6, we n repetedly use the following two opertions to find reduile onfigurtion: T 1 T T k T 1 T 2 T 3 T k T i T j T l () Trnsitions of onfigurtions. T i T j T l () Closed set of irreduile onfigurtions. Color exhnge of (, ) Color exhnge of (, ) or (, ) Fig. 6. The trnsition digrm of onfigurtions. 1) Lol opertion - move to nother stte within the sme onfigurtion T (G): Lol opertions only involve (, ) olor exhnges, whih leve ll (, ) edges in the Petersen perfet mthing intt. We n systemtilly move (, ) vriles within their (, ) loking yles until the onfigurtion rehes reduile stte. The mximum numer of moves within the stte spe S T (G) is ounded y S T (G) = 2 τ n 1 n 2... n τo. 2) Glol opertion - trnsform T (G) into nother onfigurtion T (G): Glol opertions involve (, ) nd (, ) olor exhnges, whih will hnge the Petersen perfet mthing tht determines the onfigurtion. If onfigurtion T (G) is reduile, then it will trnsform into nother onfigurtion T (G) fter eliminting two (, ) vriles. On the other hnd, if onfigurtion T (G) is irreduile, then we n trnsform T (G) into nother onfigurtion T (G) y negting some even (, ) or (, ) yles in T (G). The proess of serhing reduile onfigurtion is rndom wlk in the entire spe of onfigurtions of the ui grph G. An irreduile onfigurtion T (G) my trnsform into reduile onfigurtion T (G) y sequene of negting (, ) or (, ) yles. For exmple, the following sequene of opertions will eliminte the two (, ) vriles in the irreduile onfigurtion T (G) shown in Fig. 7(): Move one of the (, ) vriles to the edge (, ), s shown in Fig. 7(). Negte the (, ) yle ( ), s highlighted in Fig. 7(). Negte the (, ) yle ( ), s highlighted in Fig. 7(d). The two vriles re now onneted y the (, ) pth (3 7 6 ), s shown in Fig. 7(e), nd they n e eliminted y Kempe wlk.

9 () An irreduile onfigurtion T (G). () Move n (, ) vrile to edge (, ). () Negte the highlighted (, ) yle (d) Negte the highlighted (, ) yle. (e) Two (, ) vriles re onneted y Kempe pth. Fig. 7. Trnsforming n irreduile onfigurtion into reduile onfigurtion. The ridgeless ui grph G tht nnot hve proper 3-edge oloring is ommonly referred to s snrk [16], [17]. The est known snrk is the Petersen grph G P, whih is the smllest ridgeless ui grph with no 3-edge oloring, s shown in Fig. 8(). It n e esily shown tht the onfigurtions of the Petersen grph re ll equivlent y releling verties, mening tht they re isomorphi to eh other. This unique onfigurtion T (G P ) of the Petersen grph is irreduile. Although T (G P ) hs = 100 sttes, ut they re ll isomorphi to one of the two sttes shown in Fig. 8() nd 8(e), respetively, nd neither stte is reduile. One of the irreduile sttes ξ 1 of T (G P ) is shown in Fig. 8(), in whih the two (, ) vriles re respetively ontined in two disjoint (, ) yles, nd there re no (, ) or (, ) resolution yles. The two (, ) nd (, ) exlusive hins in stte ξ 1 re displyed in Fig. 8() nd 8(d), respetively. Similrly, nother irreduile stte ξ 2 of T (G P ) is shown in Fig. 8(e). Besides the two (, ) nd (, ) exlusive hins, s displyed in Fig. 8(f) nd 8(g), respetively, the stte ξ 2 of T (G P ) hs one (, ) nonessentil resolution yle. It is esy to show tht not only re ll onfigurtions of the Petersen grph irreduile; in ft, they re isomorphi to eh other. As n exmple, negting the (, ) yle ( ) in the onfigurtion T (G p ) shown in Fig. 9(), we otin nother onfigurtion T (G p ) displyed in Fig. 9(), whih is the sme s T (G p ) y releling orresponding verties of the Petersen grph. Oviously, these onfigurtions re ll irreduile euse the two (, ) vriles n never e onneted y n (, ) Kempe pth. In 1966, Tutte onjetured tht every ridgeless ui grph tht does not ontin the Petersen grph s grph minor is 3-edge olorle, or equivlently, every snrk hs the Petersen grph s grph minor [18]. Neil Roertson nd Roin Thoms nnouned in 1996 tht they proved this onjeture [16], [19], [20], ut they did not pulish the result. If this onjeture is vlid, then the 4CT n e immeditely estlished ording to Tit s equivlent formultion.

10 () The Petersen grph. () A stte ξ 1 of the Petersen grph. () (, ) su-grph of ξ 1. (d) (, ) su-grph of ξ 1. (e) A stte ξ 2 of the Petersen grph. (f) (, ) su-grph of ξ 2. (g) (, ) su-grph of ξ 2. Fig. 8. Two irreduile sttes of onfigurtion of the Petersen grph. () A onfigurtion T (G p) of the Petersen grph. () Negte (, ) resolution yle. () Re-rrnge to nother onfigurtion T (G p). Fig. 9. Isomorphi onfigurtion of the Petersen grph. Tutte s onjeture holds for lmost ll known snrks. The ontrtion proesses of some well-known snrks re illustrted in Appendix A. The Petersen grph s grph minor is not proper hrteriztion of snrks. It is esy to show tht mny 3-edge olorle grphs lso hve the Petersen grph s grph minor. For exmple, the 3-edge olorle ui grph shown in Fig. 10 is otined y dding the edge e 11,12 = (1, 2 ) to the Petersen grph, whih ertinly hs the Petersen grph s grph minor. Sine inresing the numer of vriles is prohiited in ny trnsformtion of onfigurtion, it is possile tht set of irreduile onfigurtions is losed in trnsition digrm under the trnsformtions defined ove, s illustrte in Fig. 6(), in whih n irreduile onfigurtion n only trnsform into other irreduile onfigurtions.

11 () Add n edge to the Petersen grph. () A 3-edge oloring of the modified Petersen grph. Fig. 10. A 3-edge olorle ui grph with the Petersen grph s grph minor. Therefore, n immedite onsequene is the hrteriztion of snrks given in the following theorem. Theorem 3. A ridgeless ui grph G(V, E) is Clss 2 grph if nd only if G hs losed set of irreduile onfigurtions. Note tht, for the sme reson stted ove, these irreduile onfigurtions in the losed set of the snrk should ll possess the sme minimum numer of vriles. V. PETERSEN CONFIGURATION For edge oloring of ridgeless ui plnr grphs, we re interested in prtiulr onfigurtion P (G), referred to s the Petersen onfigurtion, whih stisfies the following onditions: 1) The onfigurtion P (G) ontins two (, ) vriles. 2) The two (, ) vriles re on the oundry of pentgon in some stte ξ of P (G). It is esy to show tht the ove onditions imply tht three edges of the pentgon re ontined in the two odd (, ) Tit yles, nd the remining two (, ) edges of the pentgon elong to the perfet mthing. The stte ξ of P (G) shown in Fig. 11 stisfies oth onditions. Fig. 11. A prtiulr stte ξ of Petersen onfigurtion P (G). The two (, ) vriles in Petersen onfigurtion P (G) n e trnsformed into two (, ) vriles y olor exhnges performed t the two verties nd, s shown in Fig. 12(). The result is displyed in Fig. 12(), in whih the two (, ) vriles re ontined in two disjoint (, ) yles; otherwise, they n e esily neled y Kempe wlk. Finlly, the omplete stte ξ of Petersen onfigurtion P (G) under onsidertion is depited in Fig. 12(). If we perform (, ) olor exhnge opertions t verties, nd, s shown in Fig. 13(), the result is displyed in Fig. 13(), in whih the right (, ) hin eomes n (, ) exlusive hin tht onnets the two (, ) vriles, nd the left (, ) hin eomes prt of the (, ) resolution yle ( ) tht inludes ll verties,,,,, nd, of the pentgon. If this (, ) resolution yle is essentil, then the two (, ) vriles n e neled, nd the Petersen onfigurtion P (G) is reduile. On the other hnd, if this (, ) resolution yle is nonessentil, then we n test other sttes of this Petersen onfigurtion y moving (, ) vriles or negting (, ) yles.

12 () Two disjoint (, ) yles. () Two disjoint (, ) yles. () The omplete stte ξ. Fig. 12. The omplete stte of Petersen onfigurtion P (G). () (, ) olor exhnges. () (, ) resolution yle nd (, ) exlusive hin. () The stte ξ of the Petersen grph. Fig. 13. The Tit yle struture of Petersen onfigurtion P (G). A Petersen onfigurtion P (G) is irreduile if ll sttes of P (G) re irreduile. The onfigurtion of the Petersen grph shown in Fig. 13() is the smllest irreduile Petersen onfigurtion P (G). For plnr Petersen onfigurtions, we hve experimentlly tested over one hundred thousnd instnes generted y omputer, nd found tht they re ll reduile. Sine ny two-olored pths of the sme kind nnot ross eh other in the plne, it is evident y our pereption tht n irreduile Petersen onfigurtion P (G) with the Tit yle struture shown in Fig. 13() must e non-plnr. Therefore, we propose the following postulte: Reduiility Postulte. Every Petersen onfigurtion P (G) of ridgeless ui plnr grph G(V, E) is reduile. Despite the ft tht this proposition hs een verified y more thn one hundred thousnd instnes, we still don t hve logil proof of this ssertion. We oserved the following properties of Petersen onfigurtion from experimentl results: 1) The rdinlity of the stte spe S P (G) of Petersen onfigurtion P (G) is given y S P (G) = 4 2 τe n 1 n 2, where τ e is the numer of even (, ) yles, nd n 1 nd n 2 re the respetive numers of verties in the two odd (, ) yles. In experimentl testing of the reduiility of the Petersen onfigurtion, we did not onsider the negtion of even (, ) yles to simplify the omputtionl omplexity. Yet we were still le to find reduile sttes in the redued stte spe of size 4n 1 n 2. However, in theory, it is not ler whether we n lwys ignore the negtion of even (, ) yles. 2) For eh Petersen onfigurtion P (G), there is ompnion onfigurtion P (G) tht ontins two (, ) vriles, s shown in Fig. 12(). Thus, the onfigurtion P (G) n e onsidered s reduile if ny stte of P (G) is reduile. In our experimentl testing, however, we hve never enountered ny se in whih the onfigurtion P (G) is irreduile ut the ompnion onfigurtion P (G) is reduile. Furthermore, our experimentl results show tht there re usully mny reduile sttes in Petersen onfigurtion P (G) of ridgeless ui plnr grph G. As n exmple, reduile P (G) of the plnr grph G is shown in Fig. 14. The pentgon ( ) of grph G is highlighted in Fig. 14(), in whih the two (, ) vriles re loted t edge (, 3 ) nd (2, 9 ), respetively. The two loking (, ) yles with length 9 nd 21, respetively, re displyed in Fig. 14(). This onfigurtion P (G) hs = 756 sttes, mong whih 284 of them re reduile.

13 v () A Petersen onfigurtion. () Two highlighted (, ) loking yles. Fig. 14. A Petersen onfigurtion P (G) with 284 reduile sttes. It should e noted tht the seond ondition of Petersen onfigurtion P (G) is neessry in the postulte of reduiility, whih requires tht oth odd (, ) yles of P (G) ontin the oundry edges of the sme pentgon. As n exmple, the onfigurtion T (G) shown in Fig. 7() hs two odd (, ) yles, eh of length 5, nd two even (, ) yles. The two odd (, ) yles ontin oundry edges of two different pentgons; therefore T (G) does not stisfy the seond ondition of Petersen onfigurtion. We hve heked tht the = 400 sttes of this onfigurtion T (G) re ll irreduile. VI. THREE-EDGE COLORING THEOREM In this setion, we provide n lgorithmi pproh to prove tht every ridgeless ui plnr grph G(V, E) is 3-edge olorle. The si ide is to reursively olor the ui grph y indution on the numer of verties V. Initilly, it is trivil to show tht the smllest simple ui grph G with V = 4 nd E = 6 is 3-edge olorle. For ridgeless ui plnr grph G with V = n, we prove tht G is 3-edge olorle if the ui grph G otined y deleting n edge in G is 3-edge olorle. The indution steps involve two opertions, edge deletion nd edge insertion, defined s follows. An edge-deletion opertion is performed on n unolored ui grph G. As shown in Fig. 15(), new ui grph G with V 2 verties nd E 3 edges n e otined from G(V, E) y deleting n edge e nd smoothing out the two end nodes with degree 2. An edge-insertion opertion shown in Fig. 15() is performed on olored onfigurtion T (G) of the ui grph G. The insertion opertion will introdue two new vriles t the two ends of the inserted edge. delete smooth () Illustrtion of edge deletion. dd nodes join () Illustrtion of edge insertion. Fig. 15. Illustrtions of edge deletion nd edge insertion.

14 In grph theory, the girth of grph is the length of the shortest yle ontined in the grph. The girth of plnr grph G is the minimum numer of edges surrounding fe in G. We need the following two lemms in the indutive steps of our lgorithm. Lemm 1. The girth of ridgeless ui plnr grph G(V, E) is less thn or equl to 5. Proof: Suppose the ui grph G(V, E) emedded on sphere hs f fes, v verties, nd e edges. From Euler s formul, we hve v e + f = 2. (5) Suppose the girth of G is lrger thn 5, then we hve 6f 2e. (6) In ui grph G(V, E), we know tht 3v = 2e. From (5), the ove inequlity (6) implies 6f 6f 12, whih is impossile. An edge e in ridgeless ui plnr grph G(V, E) is n dmissile edge, if the grph G otined y deleting edge e remins ridgeless. The following lemm oviously holds in ny ridgeless ui plnr grph. Lemm 2. Any fe of ridgeless ui plnr grph G(V, E) hs t lest one dmissile edge. In the rest of this setion, we show tht the following 3-edge oloring theorem is n immedite onsequene of the postulte of reduiility of the Petersen onfigurtion. Theorem 4. Every ridgeless ui plnr grph G(V, E) hs 3-edge oloring. Proof: As we mentioned ove, the 3-edge oloring of ridgeless ui plnr grph G(V, E) with V = n n e derived from the oloring of the ui grph G, whih is otined y deleting n edge in G. Thus, our indution step strts with the seletion of fe F in grph G with the minimum numer of oundry edges, whih is less thn 6 ording to Lemm 1. The fe F hs n dmissile edge e ording to Lemm 2, suh tht the ui grph G, derived from G y deleting edge e, is still ridgeless. Suppose tht ny ridgeless ui plnr grphs with V = n 2 verties re 3-edge olorle. We prove tht grph G is 3-edge olorle y onsidering ll possile girths of grph G. First, it is trivil to show tht the indution is vlid if the girth is equl to 2. The remining three nontrivil ses re desried s follows. 1) The girth of G equls 3. Suppose the ui grph G(V, E) with n verties hs fe F (,, ) with three oundry edges s shown in Fig. 16(). We ssume, without loss of generlity, tht edge e 3,4 = (, ) is dmissile. Then ridgeless ui grph G with n 2 verties n e otined y deleting the edge e 3,4 nd smoothing out the two end verties nd. By indution hypothesis, the ui grph G hs 3-edge oloring. Suppose the two edges e 2,5 = (, ) nd e 2,6 = (, ) in G re olored y nd, respetively, s shown in Fig. 16(). Then we n derive 3-edge oloring of grph G s shown in Fig. 16(d). 2) The girth of G equls 4. Suppose the ui grph G(V, E) with n verties hs fe F (,,, ) with four oundry edges s shown in Fig. 17(). We ssume tht edge e 3,4 = (, ) is dmissile. As efore, ridgeless ui grph G with n 2 verties n e otined y deleting the edge e 3,4 nd the two end verties nd. By indution hypothesis, the ui grph G hs 3-edge oloring. Suppose the olor is ssigned to edge e 1,2 = (, ) in G. Then the oloring of edge e 3,4 = (, ) in G is determined y the oloring of edge e 1,8 = (, ) nd e 2,7 = (, ) in G. We onsider the following two su-ses: ) If edge e 1,8 nd e 2,7 re olored with sme olor, sy olor, s shown in Fig. 17(). Then we n ssign olor to the inserted edge e 3,4 = (, ), s shown in Fig. 17(). The two (, ) vriles reted y inserting edge e 3,4 n e eliminted y Kempe wlk long the pth ( ),

15 () A tringle in G. () A 3-edge oloring of G. () A 3-edge oloring of G. (d) A proper 3-edge oloring of G. Fig. 16. Reursive 3-edge oloring of G when girth is 3. y performing olor exhnges t nd. We otin proper 3-edge oloring of G, s shown in Fig. 17(d), fter eliminting these two (, ) vriles. ) If edge e 1,8 nd e 2,7 re olored with different olors, then we onsider the following two irumstnes: i) As shown in Fig. 17(e), the two (, ) pths ( ) nd ( ) re ontined in different (, ) yles. Negte one of the (, ) yle in G, sy the one tht ontins the edge e 2,7. As result, the two edges e 1,8 nd e 2,7 hve the sme olor s shown in Fig. 17(f). Then we n esily derive 3-edge oloring of G, s shown in Fig. 17(g), in wy similr to su-se 2 desried ove. ii) As shown in Fig. 17(h), the two (, ) pths ( ) nd ( ) re ontined in the sme (, ) yle. We first insert edge e 3,4 = (, ) into G. The inserted (, ) edge e 3,4 introdues two (, ) vriles t the two ends to mke the oloring of G onsistent, s shown in Fig. 17(i). Sine these two vriles re ontined in the sme (, ) yle, they n e esily eliminted y Kempe wlk, s shown in Fig. 17(i). The resulting 3-edge oloring of G is exhiited in Fig. 17(j). 3) The girth of G equls 5. Suppose the ui grph G(V, E) with n verties hs fe F (,,,, ) with five oundry edges s shown in Fig. 18(). We ssume tht edge e 1,2 = (, ) is dmissile. We n otin grph G with n 2 nodes y deleting edge e 1,2 nd smoothing out the two end verties nd, s shown in Fig. 18(). By indution hypothesis, the ui grph G hs n 3-edge oloring. The two edges e 5,8 nd e 3,6 in G my hve the sme olor, sy olor, s shown in Fig. 18(), or different olors, sy olor nd, s shown in Fig. 18(d). In oth ses, two (, ) vriles will e reted if we insert edge e 1,2 k into G, s shown in Fig. 18() nd 18(e), respetively. If the two vriles re ontined in the sme (, ) yle, then they n e eliminted y simple Kempe wlk. Otherwise, grph G hs Petersen onfigurtion P (G) if these two vriles re ontined in different (, ) yles. Grph G is 3-edge olorle euse the Petersen onfigurtion P (G) is reduile ording to the reduiility postulte.

16 () A qudrngle in G. () A 3-edge oloring of G. (e) A 3-edge oloring of G. (h) A 3-edge oloring of G. () A 3-edge oloring of G. (f) Coloring of G fter negting one (, ) yle. (i) A 3-edge oloring of G. (d) A proper 3-edge oloring of G. v 8 (g) A proper 3-edge oloring of G. v 6 v 8 (j) A proper 3-edge oloring of G. Fig. 17. Reursive 3-edge oloring of G when girth is 4. () A pentgon in G. () A 3-edge oloring of G. () A 3-edge oloring of G. (d) A 3-edge oloring of G. (e) A 3-edge oloring of G. Fig. 18. Reursive 3-edge oloring of G when girth is 5.

17 Remrk: The 4CT is therefore estlished ording to Tit s equivlent formultion mentioned in Setion I. It should e noted tht euse the reduiility postulte requires tht the Petersen onfigurtion n e redued y single essentil yle, the 4CT does not imply tht every Petersen onfigurtion hs this property. Therefore, the existing omputer-ssisted proof of 4CT does not pply to the proof of reduiility postulte. VII. COMPARISONS AND DISCUSSIONS The reduiility postulte of the Petersen onfigurtion in grph theory nd the prllel postulte in Euliden geometry shre some ommon hrteristis of the two-dimensionl plne. A omprison etween these two propositions is desried s follows. Invrints of the plne. In the two-dimensionl plne, the well-known geometri invrint tht the sum of the ngles in every tringle equls π is n immedite onsequene of the prllel postulte. Similrly, in the lst setion, we proved tht the reduiility postulte of the Petersen onfigurtion implies tht the hromti index of ridgeless ui plnr grphs equls 3, or equivlently, the minimum numer of olors to olor geogrphil mp is 4, whih is topologil invrint of the plne. Solvility onditions of two equtions in the plne. In Eulids Elements, postulte 5, the originl version of the prllel postulte, is stted s follows: If stright line flling on two stright lines mkes the interior ngles on the sme side less thn two right ngles, the two stright lines, if produed indefinitely, meet on tht side on whih re the ngles less thn the two right ngles. Consider eh line s liner eqution. The prllel postulte is tully the solvility ondition of two liner equtions in the plne, whih is somewht nlogous to tht of the reduiility postulte. In ridgeless ui plnr grph G, the reduiility postulte lims tht every Petersen onfigurtion P (G) of G is reduile, or equivlently, the two (, ) vriles of P (G) n e onneted y n (, ) Kempe pth in some stte of P (G), vi negting resolution yle. The ommon hrteristis of the prllel postulte nd the reduiility postulte re listed in Tle II. TABLE II A COMPARISON BETWEEN PARALLEL POSTULATE AND POSTULATE OF REDUCIBILITY Prllel Postulte In the plne Two lines interseted y third line Sum of inner ngles < π Reduiility Postulte In the plne Two loking yles joined y n essentil resolution yle Bounded y pentgon The nlogy etween these two solvility propositions in the plne is illustrted in Fig. 19. The prllel postulte sserts the solvility ondition of two lines y using third line, s highlighted in Fig. 19(), tht intersets oth lines, while the reduiility postulte sserts the solvility of two odd (, ) yles y using n essentil (, ) resolution yle, s highlighted in Fig. 19(), tht intersets oth loking yles. The two (, ) loking yles nd n essentil (, ) resolution yle tht ompose the Petersen onfigurtion re shown in Fig. 19() nd Fig. 19(d), respetively. Furthermore, the ounterprt of the ondition θ 1 +θ 2 < π is tht oth (, ) loking yles must ontin the edges of fe with less thn six order edges. Note tht the oundry of the entire onfigurtion shown in Fig. 19() is pentgon. These two similr ounding onditions represent some extruding onstrints tht ensure the existene of solution in the plne. On the other hnd, the Petersen onfigurtion my e irreduile if the ui grph is non-plnr. As we illustrted in Fig. 9 efore, ll onfigurtions of the Petersen grph re irreduile nd they re isomorphi to eh other. The ove nlogy etween the prllel postulte nd the reduiility postulte not only revels the signifine of the onept of omplex oloring in grph theory, ut lso hints tht logil proof of the proposition is most likely impossile, the sme s prllel postulte.

18 θ 1 θ 2 () Prllel postulte. () Reduiility postulte. () Two (, ) loking yles. (d) (, ) essentil resolution yle. Fig. 19. Anlogy etween prllel postulte nd reduiility postulte. VIII. CONCLUSIONS As we mentioned efore, the onfigurtion of ridgeless ui grph G is uniquely determined y Petersen s perfet mthing. Lovász nd Plummer onjetured tht the numer of perfet mthings ontined in G is exponentil to the numer of verties of G in [23]. The onjeture ws settled y Esperet et l. in [24]. The identifition of snrk G with n verties is determined y the numer of non-isomorphi onfigurtions, denoted s θ(g), in the losed set of irreduile onfigurtions of G, nd the numer of vriles in eh onfigurtion, lled oddness nd denoted s κ(g) [25]. It hs een shown in [26] tht the oddness κ(g) is unounded. Presumly, the omplexity of deiding whether ui grph is snrk should e t lest on the order of O(θn κ ), even exluding the testing of isomorphism. Therefore, it is not fesile to dedue n effiient deterministi lgorithm to identify snrks. On the other hnd, s snrks re lerly speified, the development of some effiient rndomized lgorithms is possile in the future. In respet to omputtion omplexity, the postulte of reduiility is onsistent with the omputer-ssisted proof of the 4CT. Despite the ft tht it is NP-omplete to determine the hromti index of n ritrry ui grph, this prolem for plnr grphs n e solved in polynomil time euse the omputer-ssisted proof of the 4CT tully gives qudrti lgorithm for mp oloring, s desried in [27]. The proof of the 3-edge oloring theorem presented in Setion VI lso implies polynomil time lgorithm. The onsisteny of the omplexities of these two drstilly different pprohes suggests tht the reduility postulte of the Petersen onfigurtion should hold in every ridgeless ui plnr grph. The omplexities for edge oloring of ridgeless ui grphs re lssified in Fig. 20, sed on the ssumption tht the postulte of reduiility is vlid. Suppose, y ontrst, tht some Petersen onfigurtions of ridgeless ui plnr grphs re irreduile, suh tht the postulte of reduiility is invlid. Then the omplexity of edge oloring of plnr ui grphs would lso e in the lss of NP-omplete, the sme s non-plnr ses, euse it would involve serhing for reduile stte in the entire spe of ll onfigurtions. This senrio onflits with the qudrti lgorithm for mp oloring derived from the omputer-ssisted proof of the 4CT.

19 Bridgeless ui grph Non-plnr Plnr Clss 2 A losed set of irreduile onfigurtions. Clss 1 Trnsform irreduile onfigurtion into reduile onfigurtion. Clss 1 Every Petersen onfigurtion is reduile. Clss NP-omplete: Glol serhing for reduile stte in the spe of onfigurtions. Clss P: Lol serhing for reduile stte in Petersen onfigurtion. Fig. 20. The lssifition of ridgeless ui grphs. Another NP-omplete prolem tht ould e tkled y omplex olor exhnges is finding Hmiltonin yles [22]. A simple grph G my hve more thn one proper olor onfigurtions, nd onfigurtion n e trnsformed into other onfigurtions y negting mximl two-olored Tit yles. A onfigurtion is Hmiltonin if it ontins two-olored Hmiltonin yle. By rndom wlks on the entire spe of onfigurtions, solution of given grph G ould e rehed if G is Hmiltonin. This is only n exmple to show tht the pplition of omplex oloring to solve some hrd omintoril prolems ould e hllenging reserh topi in the future. ACKNOWLEDGMENT Tony T. Lee ws supported y the Ntionl Siene Foundtion of Chin under Grnt Qingqi Shi ws supported y the Hong Kong RGC Ermrked Grnt CUHK REFERENCES [1] A. B. Kempe, On the geogrphil prolem of the four olours, Amerin Journl of Mthemtis, vol. 2, no. 3, pp. pp , [2] P. G. Tit, Note on theorem in geometry of position, Trns. Roy. So. Edinurgh, vol. 29, pp , [3] K. Appel nd W. Hken, Every plnr mp is four olorle. Prt I. Dishrging, Illinois J. Mth, vol. 21, pp , [4], Every plnr mp is four olorle. Prt II: Reduiility, Illinois J. Mth, vol. 21, pp , [5] N. Roertson, D. Snders, P. Seymour, nd R. Thoms, The four-olour theorem, Journl of Comintoril Theory, Series B, vol. 70, no. 1, pp. 2 44, [6] R. Fritsh nd G. Fritsh, The Four Color Theorem: History, Topologil Foundtions, nd Ide of Proof, ser. Springer finne. Springer Verlg Gmh, [7] R. Wilson, Four Colors Suffie: How The Mp Prolem Ws Solved, ser. Prineton Pperks. Prineton University Press, [8] R. Thoms, An updte on the four-olor theorem, Noties Amer. mth. So, vol. 45, pp , [9] J. Petersen, Die Theorie der regulören Grphen, At Mthemti, vol. 15, pp , [10] G. Chrtrnd, L. Lesnik, nd P. Zhng, Grphs nd Digrphs, ser. A Chpmn & Hll ook. Tylor & Frnis, [11] D. West, Introdution to grph theory. Prentie Hll, [12] V. Vizing, On n estimte of the hromti lss of p-grph, Diskret. Anliz, vol. 3, pp , [13] T. Lee, Y. Wn, nd H. Gun, Rndomized -edge olouring vi exhnges of omplex olours, Int. J. Comput. Mth., vol. 90, no. 2, pp , Fe [14] J. Petersen, Sur le théorème de tit, L Intermédiire des Mthémtiiens, vol. 5, pp , [15] M. Greenerg, Euliden nd Non-Euliden Geometries: Development nd History. W. H. Freemn, [16] S.-M. Belstro, The ontinuing sg of snrks, The College Mthemtis Journl, vol. 43, no. 1, pp. pp , [17] D. Holton nd J. Sheehn, The Petersen Grph, ser. Austrlin Mthemtil Soiety Leture Series. Cmridge University Press, 1993, no. 7. [18] W. Tutte, On the lgeri theory of grph olorings, Journl of Comintoril Theory, vol. 1, no. 1, pp , [19] N. Roertson, P. Seymour, nd R. Thoms, Tutte s edge-olouring onjeture, Journl of Comintoril Theory, Series B, vol. 70, no. 1, pp , [20] R. Thoms, Reent exluded minor theorems for grphs, in In surveys in Comintoris. Univ. Press, 1999, pp [21] I. Holyer, The NP-ompleteness of edge-oloring, SIAM Journl on Computing, vol. 10, no. 4, pp , 1981.

20 [22] C. Ppdimitriou, Computtionl Complexity, ser. Mthemtis / seond level ourse. Addison-Wesley, 1995, no. v [23] L. Lovász nd M. D. Plummer, Mthing theory, Annls of Disrete Mthemtis, vol. 29, [24] L. Esperet, F. Krdo s, A. D. King, D. Král, nd S. Norine, Exponentilly mny perfet mthings in ui grphs, Advnes in Mthemtis, vol. 227, no. 4, pp , [25] A. Huk nd M. Kohol, Five yle doule overs of some ui grphs, Journl of Comintoril Theory, Series B, vol. 64, no. 1, pp , [26] E. Steffen, Mesurements of edge-unolorility, Disrete Mthemtis, vol. 280, no. 1-3, pp , [27] N. Roertson, D. P. Snders, P. Seymour, nd R. Thoms, Effiiently four-oloring plnr grphs, in Proeedings of the twenty-eighth nnul ACM symposium on Theory of omputing, ser. STOC 96. New York, USA: ACM, 1996, pp APPENDIX The ontrtion of snrk with two (, ) vriles to the Petersen grph n e hieved y the following proedure: 1) Delete ll (, ) edges tht re not ontined in the two odd (, ) yles, nd smooth out ll degree two verties on the (, ) hins, whih eome (, ) edges. 2) Delete ll internl hords of the two odd (, ) yles, nd smooth out ll degree two verties on these two yles. 3) Find suset of five externls hords, the (, ) edges tht onnet the two (, ) yles, tht forms the sme onnetion pttern s the five externl hords in the onfigurtion T (G P ) of the Petersen grph, s shown in Fig. 9(). In the onfigurtion desried ove, the term externl hord refers to the (, ) edge tht onnets the two disjoint (, ) yles, nd the term internl hord refers to the (, ) edge with oth ends terminted on the sme (, ) yle. The ontrtion of severl snrks is shown in Fig. 21 through Fig () Flower snrk. () A onfigurtion of Flower snrk. () Find 5 hords mthed with T (G p) of the Petersen grph (d) Redue to the Petersen grph. (e) A sudivision of the Petersen grph emedded in Flower snrk. Fig. 21. Contrtion of Flower snrk to the Petersen grph.