Extending precolorings of subgraphs of locally planar graphs

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1 European Journal of Combnatorcs 25 (2004) Extendng precolorngs of subgraphs of locally planar graphs Mchael O. Albertson a,joanp.hutchnson b a Department of Mathematcs, Smth College, Northampton, MA 01063, USA b Department of Mathematcs and Computer Scence, Macalester College, St. Paul, MN 55105, USA Receved 20 September 2002; accepted 18 June 2003 Avalable onlne 23 January 2004 Abstract The wdth of an embedded graph s the length of a shortest noncontractble cycle. Suppose G s embedded n a surface S (ether orentable or not) wth large wdth. In ths case G s sad to be locally planar. Suppose P V (G) s aset of vertces such that the components of G[P] are each 2-colorable, have bounded dameter and are sutably dstant from each other. We show that any 5-colorng of G[ P] n whch each component s 2-colored extends to a 5-colorng of all of G. Thus, for an arbtrary surface, the extenson theorems for precolorngs of subgraphs of locally planar graphs parallel the results for planar graphs. Crucal to the proof of ths result s the nce cycle lemma, vz. If C s a mnmal, noncontractble, and nonseparatng cycle n a so-called orderly trangulaton of at least moderate wdth, then there s a cycle C such that C les wthn the fourth neghborhood of C, C s mnmal, homotopc to C, andc ether has even length or contans a vertex of degree 4. Such a nce cycle s useful n producng 5-colorngs. We ntroduce the dea of optmal shortcuts n order to prove the nce cycle lemma and the dea of relatve wdth n order to prove the man theorem. Our results generalze to extenson theorems for precolorngs wth q 3colors Elsever Ltd. All rghts reserved. 1. Introducton Suppose G s an r-colorable graph and P V (= V (G)) s r-precolored. It s natural to ask whether the precolorng extends to a t-colorng of the entre graph where r t [11, 12]. The answer depends on the context. For example, a 4-precolorng of some vertces of a planar graph need not extend to a 4-colorng of the entre graph even f there are only two E-mal addresses: albertson@smth.edu (M.O. Albertson), hutchnson@macalester.edu (J.P. Hutchnson) /$ - see front matter 2003 Elsever Ltd. All rghts reserved. do: /j.ejc

2 864 M.O. Albertson, J.P. Hutchnson / European Journal of Combnatorcs 25 (2004) precolored vertces and they are far apart. In contrast, a 5-precolorng of sutably separated vertces extends to a 5-colorng of any planar graph [1]. It s no surprse that the queston of whether a precolorng extends s NP-complete even assumng severe restrctons on G.Itscomplextyhas been well studed; see [14]. Stll there are general results. We ntroduce a lttle notatonbefore descrbng some of what s known. The chromatc number of G, denoted by χ(g), sthemnmum number of colors needed to properly color the vertces of G.WeletG[P] denote the subgraph of G nduced by P. P s called ndependent f G[P] contans no edges. It s convenent to partton P = P 1 P 2 P k V where P 1, P 2,...,P k nduce the connected components of G[P]. Wesaythat d(p) ρ f for every par of vertces x P, y P j ( j) the dstance (number of edges n a shortest path) between x and y s at least ρ. Thestory begns wth a precolored ndependent set. Theorem 1 ([1]). If χ(g) = r, P sndependent, and d(p) 4,thenany (r+1)-colorng of G[P] extends to an (r + 1)-colorng of all of G. Ths generalzes to extendng precolorngs of q-colorable subgraphs. Theorem 2 ([3]). Suppose χ(g) r and P Vssuchthat χ(g[ P]) q and d(p) 4.Any(r +q)-colorng of G[P] n whch each G[P ] s q-colored (not necessarly wth the same colors) extends to an (r + q)-colorng of G. The precedng results are best possble both wth respect to the dstance constrant and the number of colors necessary. If we assume that G s planar, we can save a color. Note that the followng theorem s not true for any dstance constrant when q = 1[1]. Theorem 3 ([3]). Suppose G has no K 5 mnor, χ(g) r, and P V s suchthat d(p) 8 and χ(g[p]) qwhereq 2. Any(r + q 1)-colorng of G[P] n whch each G[P ] s q-colored (not necessarly wth the same colors) extends to an (r + q 1)- colorng of G. It s natural to wonder f Theorem 3 extends to the class of locally planar graphs. Our prncpal result s that ths s ndeed so. For a surface S, letɛ = ɛ(s) denote ts Euler characterstc, and let the Euler genus be gven by g = 2 ɛ. WhenG s embedded n a nonplanar surface, ts wdth w(g) s the length of a shortest noncontractble cycle [4]. Ths hasalsobeen called the edge-wdth and s known to be a crucal embeddng parameter; see [8,Chapter 5]. A graph G embedded n asurface S s sad to be locally planar f w(g) s large enough. Here large enough wll depend on S as well as the property beng sought. Theorem 3 and Thomassen s landmark Theorem 4 provde the orgn for our work. The latter also llustrates the mportance of wdth. Theorem 4 ([13]). Let G be embedded n S. If w(g) s large enough, then G can be 5-colored. We wll prove two 5-colorng extenson theorems for locally planar graphs. Wthn the ntroducton we hde detals. Our frst result, assumng a precolored ndependent set, s stated and proved n Secton 3; the second, assumng a precolored bpartte subgraph, flls Secton 4.

3 M.O. Albertson, J.P. Hutchnson / European Journal of Combnatorcs 25 (2004) Theorem 5. Let G be embedded n a surface S of Euler genus g > 0 such that w(g) s large enough. If P Vsanndependent set such that d(p) s large enough, then any 5-colorng of G[P] extends to a 5-colorng of G. Ideally we would lke a 5-colorng extenson theorem when G[P] s q-colored for q < 5 and d(p) s sutably bounded. If q 3, there s no such theorem [3]. To obtan a result for q = 2weneed to control the embeddng a bt more. For P V we ntroduce the relatve wdth of G, denoted by w P (G). Ths s the length of a shortest noncontractble cycle n whch the vertces of P do not count. Formally w P (G) = mn{ X :X V and G[X P] contans a noncontractble cycle}. Theorem 6. Suppose G s embedded n a surface S of Euler genus g > 0 and P Vs such that for each component the dameter of G[P ] s bounded. If both d(p) and w P (G) are large enough, then any 5-colorng of G[P] n whcheach component s 2-colored (not necessarly wth the same colors) extends to a 5-colorng of G. It s mmedate that 0 w P (G) w(g). Thewdth of an embedded graph can be determned n polynomal tme [8]. Determnng the relatve wdth s the same as determnng the wdth of the graph obtaned from G by contractng each component of G[P]. IfG[P] s noncontractble on the surface, the Euler genus of the contracted graph wll decrease (wth the graph possbly now on a pnched surface). Snce ths s easy to detect both the wdth and the relatve wdth are accessble parameters. Both of our precolorng extenson theorems requre the nce cycle lemma. Although ths has become part of the folklore of topologcal graph theory, t seems trcky to prove. To the best of our knowledge the proof we gve below s the frst publshed argument that works for surfaces wth g > 2. We devote Secton 2 to developng ths mportant techncal result and ntroduce the dea of optmal k-shortcuts for our proof. We antcpate that both relatve wdth and optmal shortcut wll have further applcaton n topologcal graph theory. 2. The nce cycle lemma The drvng force behnd our results and many of ther predecessors s the nce cycle lemma.thssaysthatfc s a noncontractble cycle n an orderly trangulaton (defned below) then there s a mnmal cycle C that s near to C, homotopc to C, andpartcularly nce from the pont of vew of 5-colorng. A mnmal (.e., chordless) noncontractble, and nonseparatng cycle C n an embedded graph s sad to be nce f t ether has even length or contans a vertex of degree at most 4. The frst publshed verson of a nce cycle lemma appeared n [5]. There the cycle C had to be a mnmum length noncontractble cycle n an orderly trangulaton of the torus, and the cycle C was found n the frst neghborhood of C. Stromqust establshed a nce cycle lemma for an arbtrary surface where C was found n the flled frst neghborhood of C [10], but ths was never publshed. The flled frst neghborhood mght contan vertces arbtrarly far from C. Thomassen constructed a nce cycle n [13]; however, to obtanhsverson of C he used the noton of weak geodesc, requrng a consderable detour away from C. Usng local modfcatons he found hs nce

4 866 M.O. Albertson, J.P. Hutchnson / European Journal of Combnatorcs 25 (2004) cycle on one sde of C, but,aswenow know, fndng a nce cycle requres zgzaggng to both sdes of C.Wesketchtheargument at the end of ths secton. Here we ntroduce optmal k-shortcuts and use ths dea to gve a smple proof of the lemma. Our result also has theadvantagethatc s wthn dstance four of the orgnal C. We begn wth some necessary background. If U V,letN(U) denote the set of vertces that are not n U butare adjacent to at least one vertex n U.Inductvely N (U) denotes the set of vertces that are not n N 1 (U) but are adjacent to at least one vertex n N 1 (U). Thus N (U) conssts of those vertces n G whose dstance from U s exactly. Gven a traversal of C, a2-sded cycle wthn a graph G embedded n a surface S, we defne R(C) and L(C), respectvely the (not necessarly dsjont) rght and left neghbors of C. R(C) (resp., L(C)) sthesetofvertces of N(C) that are adjacent to C along an edge emanatng from the rght (resp., left) sde of C.Allseparatng cycles of G are 2-sded as are all noncontractble cycles on an orentable surface. A nonorentable surface may contan 1-sded or 2-sded noncontractble and nonseparatng cycles. For G embedded n a surface S of Euler genus g > 0, a set of noncontractble cycles C 1, C 2,...,C s where s g s called a planarzng set f G C s a planar graph. For j, dst(c, C j ) s the length of a shortest path jonng these cycles. When C s a2-sded noncontractble and nonseparatng cycle, dst(c, C ) s the length of ashortest path begnnng wth an edge from C to L(C ) and endng wth an edge from R(C ) to C. When C s a1-sded noncontractble cycle, t s convenent to defne dst(c, C ) as nfnty. In ths case we (locally) defne R and L along M, asubpath of C that contans m edges. Suppose C s a(1-sded or 2-sded) noncontractble cycle of length at least m + 1. On a traversal of M C there are rght neghbors R(M) and left neghbors L(M), both subsets of N(C ),andthesetwosets are dsjont when m w(g) 3. Formally when m w(g) 3, R(M) ={y N(C ):theresavertexx M and an edge xy lyng on the rght sde of M n the traversal}. L(M) s defned n the same manner and gven the wdth hypothess, L(M) R(M) =. We defne R j (M) to be the vertces of N j (C ) that can be reached by a path startng n M wth an edge of R(M). WedefneL j (M) n an analogous way and note that for 2 j w(g) m 1 2, L j (M) R j (M) =.Foravertexx C we defne R(x) to be ts rght neghbors wth respect to a traversal of a (short) subpath contanng x and deg R (x) = R(x) s called the rght degree of x.wemake the analogous defnton for the left degree and note that deg(x) = deg R (x) + deg L (x) + 2. If G s atrangulaton of the surface S and U V, G[U] s sad to be orderly f every contractble 3-cycle n G[U] bounds a face of G and f every contractble 4-cycle n G[U] s ether the boundary of two trangles of G that share an edge or the frst neghbor crcut of a vertex of degree4ng. Orderly graphs have been helpful n nductve proofs of 5-colorng theorems for embedded graphs [5, 7]. Suppose C 1 = u 1, u 2,...,u t s amnmal, noncontractble, and nonseparatng cycle n a graph embedded on a surface S of Euler genus g > 0. A path of length 6, say v 1,v 2,...,v 7 s sad to be an optmal 6-shortcut for C 1 f v 1 = u for some, v 7 = u j for some j, the resultng cycle C 2 = u 1,...,u 1, {u = v 1 },v 2,...,v 6, {v 7 = u j }, u j+1,...,u t s homotopc to C 1, and C 2 s as short as possble. Note that v 2,...,v 6 may be vertces of C 1.

5 M.O. Albertson, J.P. Hutchnson / European Journal of Combnatorcs 25 (2004) Lemma 1 (Nce Cycle Lemma). Suppose C 1 s a mnmal, noncontractble, and nonseparatng cycle n a graph G, a trangulaton of a surface S of Euler genus g > 0. Suppose G C1,4 = G[C 1 N(C 1 ) N 4 (C 1 )] s orderly and w(g) 15. Theres amnmal, noncontractble, and nonseparatng cycle, say C 1,nG C 1,4 such that C 1 s homotopc to C 1 and C 1 s nce. If dst(c 1, C 1 ) d, then dst(c 1, C 1 ) d 8, and for X Vdsjont from C 1, dst(x, C 1 ) dst(x, C 1) 4. Proof. Suppose C 1 = u 1, u 2,...,u t s amnmal, noncontractble, and nonseparatng cycle n G. Choose M = v 1,v 2,...,v 7 to be an optmal 6-shortcut for C 1 where v 1 = u and v 7 = u j.the resultng cycle C 2 = u 1,...,{u = v 1 },v 2,...,v 6, {v 7 = u j }, u j+1,...,u t s mnmal snce f C 2 were to contan a chord then the 6-shortcut would not have been optmal. Snce w(g) 15, R 4 (M) L 4 (M) =.Wemayassume that both C 1 and C 2 have odd length and contan no vertex of degree 4. Thus both t and t j are odd. If D denotes the path from u j+1 to u 1 nclusve, then D = 1 + t j s even. Case (). Suppose deg R (v 3 ) 2anddeg R (v 4 ) 2. Let x denote the vertex that s n a trangle totherght of the edge jonng v 3 wth v 4.Suppose A s a shortest path from v 1 to x among the vertces n N(v 3 ) not ncludng ether endpont. Normally A ncludes v 2 but that s not necessary. Next let B be the shortest path from x to v 6 among the neghbors of v 4 not ncludng ether endpont. Normally B ncludes v 5 but that s not necessary. Consder the cycle C 3 = D,v 1, A, x,v 4,v 5,v 6,v 7. C 3 s homotopc to C 2.Letus examne the possble chords for C 3.Iftherewere a chord from D to any vertex n the subpath A, x,v 4,v 5,v 6,thenweddnot select our optmal 6-shortcut correctly. Ths s also the case f there were a chord from A to ether v 6 or v 7.Iftherewere a chord from A to v 4,thenG C1,4 contans a contractble 3-cycle that s not a face boundary. If there were achord from A to v 5,thenwehaveacontractble 4-cycle, say a,v 3,v 4,v 5,wth x n ts nteror. If x s adjacent to v 5 and deg(x) = 4, then D,v 1,v 2,v 3, x,v 5,v 6,v 7 s nce. If x s not adjacent to v 5 or deg(x) 4, then G C1,4 contans a forbdden contractble 4-cycle. Thus ether A s odd or we are done snce C 3 would be nce. Next consder the cycle C 4 = D,v 1,v 2,v 3, x, B,v 6,v 7.Asnthe precedng paragraph ether B s odd or we are done snce C 4 would be nce. Fnally, look at C 5 = D,v 1, A, x, B,v 6,v 7. C 5 s homotopc to C 2 and C 5 s even. The only chord we have not consdered s an edge from A to B. Ths wll produce a contractble 4-cycle. Ether deg(x) = 4andC 4 s nce or G C1,4 contans a forbdden contractble 4-cycle. Thus we are fnshed the proof of Case (). Case (). Suppose C 2 s our mnmal, noncontractble, and nonseparatng cycle and deg R (v 3 )<2. If ether deg R (v 3 ) = 0ordeg R (v 3 ) = deg R (v 4 ) = 1, then v 1,...,v 7 s not an optmal 6-shortcut. Thus we may assume that deg R (v 3 ) = 1anddeg R (v 4 ) 2. If deg R (v 5 ) 2, then we are back n Case () usng v 4 and v 5.Thus we may assume deg R (v 5 ) = 1. If ether deg L (v 5 ) = 1ordeg L (v 3 ) = 1, then C 2 s nce. Now f deg L (v 4 )>1, then we are back n Case () nterchangng left wth rght. Thus we may assume that deg R (v 3 ) = deg L (v 4 ) = 1anddeg L (v 5 ) 2. Suppose x s v 3 s unque rght neghbor. We alter C 2 by replacng v 3 wth x. Inthe resultng cycle both deg L (v 4 ) and deg L (v 5 ) are at least 2 and we are n Case () once agan.

6 868 M.O. Albertson, J.P. Hutchnson / European Journal of Combnatorcs 25 (2004) Case (). Suppose C 2 s our mnmal, noncontractble, and nonseparatng cycle, deg R (v 3 ) 2, and deg R (v 4 ) < 2. As n Case () we may assume deg R (v 4 ) = 1. If deg R (v 5 ) 1, then v 1,...,v 7 s not an optmal 6-shortcut. If deg R (v 5 ) 2, then by reversng the drecton of the 6-shortcut we are n Case (). In each case we fnd the nce cycle wthn G C1,4. Thus each nce cycle s no more than four edges away from ts orgnal planarzng cycle. The dstance clams follow mmedately. We dgress a moment to descrbe how to construct a noncontractble cycle C n a graph embedded n a nonorentable surface wth the property that a nearby nce cycle must use both sdes of C. LetC consst of vertces whose rght degrees are alternately 1 and 3. It s easy to buld such a cycle n a 6-regular trangulaton of a Klen bottle, and from there t can be on any nonorentable surface. For such a cycle, there s no nce replacement cycle lyng locally on one sde of C. Thus when C has odd length and s 1-sded, the nce cycle must be found usng detours locally to both sdes of C. 3. When an ndependent set s precolored Averson of the next theorem for orentable surfaces appeared n [2]. There are several good reasons to have another look. Frst, the proof gven below works for both orentable and nonorentable surfaces. Second, the partcular nce cycle lemma needed n the earler proof never appeared n prnt. Thrd, several steps n our proof are reused n Secton 4. Fnally, the constants are better. Note that Theorem 7 s a specfc realzaton of Theorem 5. Theorem 7. Suppose G s embedded n a surface S of Euler genus g > 0 and w(g) 208(2 g 1). IfP V s an ndependent set n G such that d(p) 18, thenany 5-colorng of P extends to a 5-colorng of G. Proof. We may assume G s atrangulaton of S. Otherwse we could add vertces and edges to G,makng t a trangulaton whle keepng the wdth and dstance between vertces unchanged [2, 5, 12]. Thomassen and Yu have shown that f w(g) 8(d+1)(2 g 1),then there exsts a planarzng set of cycles C 1, C 2,...,C s, s g,suchthat dst(c, C j ) 2d for 1 j s [13, 15]. For our purposes we need d = 25. If for some p P and s,dst(p, C )<9, we create a new mnmal cycle, say C,byreplacng vertces whose dstance s less than 9 from p by a path n N 9 (p).wth the gven dstance constrants, these altered C do not ntersect and do not come too close to another p P. Once we have done ths for every cycle n {C 1,...,C s } and for every p P, we have a new collecton of planarzng cycles {C 1,...,C s } wth each C homotopc to C. Snce the maxmum dstance between any two vertces wthn N 9 (p) s 18, the dstance nvolvng one cycle can be changed by at most 18. Snce dst(c, C j ) 2d = 50, we have dst(c, C j ) 2d 36 = 14 and dst(p, C ) 9for each, j s. Next, for each s we make the subgraph G C,4 orderly (see Lemma 1). For each G C,4 we delete vertces of G nteror to any contractble 3-cycle of G C,4 and, f there s more than one, all vertces nteror to any contractble 4-cycle of G C,4. Inthelatter case, we add a vertex v adjacent to all four boundary vertces. We may have deleted vertces of P, but as we shall see below, none nteror to and adjacent to vertces of a contractble

7 M.O. Albertson, J.P. Hutchnson / European Journal of Combnatorcs 25 (2004) or 4-cycle. In addton we have not changed the dstance between any two vertces of P exteror to these contractble cycles. Thus t s stll the case that d(p) 18 and dst(p, C ) 9. Call ths ntermedate graph G 1.Thenfor 1 s we apply Lemma 1 wth C = C 1 and let C denote the resultng nce cycle, homotopc to C,lyng wthn G C,4.Thsgvesthefnal planarzng set of cycles {C 1,...,C s }. We have dst(c, C j ) 2d 36 8 = 6anddst(P, C ) 5. Wth ths planarzng set for G 1 we form a planar trangulaton. For each C that s 2-sded, 1 s, we remove C and replace t wth two vertces x and y.then for each edge n L(C ), x s adjacent to both endponts of that edge, and for each edge n R(C ), y s adjacent to both endponts. For C 1-sded, we remove C and replace t wth x adjacent to both ends of each edge n N(C ).The resultng graph G 0 s aplane trangulaton wth the dstance between every par of new vertces, x, y,atleast 6. Ther dstance from P s at least 5. Suppose c s a 4-colorng of G 0 usng colors {1, 2, 3, 4} [6, 9]. If for some, c(x ) c(y ),werecolor each vertex n N 2 (y ) that s colored c(y ) wth color 5 and then at y we perform a (c(x ), c(y ))-Kempe change so that y gets the same color as x.snce dst(x, y ) 6, ths recolorng s vald. Next we transfer ths colorng back to G 1.For each s for whch C has even length, we 2-color C wth {c(x ), 5}, and for each C wth a vertex v of degree 4, we 2-color C wth {c(x ), 5} except for v whch s colored last wth whatever color s avalable. Next we correct the colorng for the vertces n P G 1 that were not colored correctly by c. Sncedst(P, C ) 5, c(p) s one of {1, 2, 3, 4}. Ifanyp P was precolored wth color 5, ts color can be changed mmedately. If p was precolored wth, say 1, and c(p) = 2, then we recolor all neghbors of p that are colored 1 wth color 5, and then gve p ts desred color 1. Due to dstance constrants, ths produces a proper colorng on G 1 ; call ths colorng c 1. Fnally, we add back and color the vertces deleted wthn contractble 3- and 4-cycles lyng wthn G C,4 for each s. LetC be a contractble 3-cycle n G C,4 wth nonempty nteror n G. Thus dst(c, C ) 4. Call the plane graph nduced by C and ts nteror H. H may contan some elements of P, necessarly at dstance at least 9 from C and so at dstance at least 5 from C. Webegnwth a 4-colorng of H that agrees wth the colorng of C from G 1.Thenwecorrect the colorng of any p P H as n the precedng paragraph. Snce dst(p, C) >2, the colorng on C s not changed and the two colorngs are consstent. Suppose C = u,v,x, y s a contractble 4-cycle wthn G C,4 that wthn G contans more than one nteror vertex. Thus some nteror vertces were deleted to form G 1.Let H be the plane graph on C and ts nteror n G. Wemayassume that n the colorng of G 1, c 1 (u) = 1andc 1 (v) = 2. Case (). Suppose c 1 (x) = 1. We transform the plane graph H by dentfyng the vertces u and x wthn the outer face of H.Ifc 1 (y) = 2 (resp., 3), then we 4-color H usng colors {1, 3, 4, 5} (resp., {1, 2, 4, 5}) makngsure that u and x (resp., u,v,andx) are correctly colored. Then we correct the colorngofanyvertces p H as above. Snce agan dst(p, C) >2, the color 2 (resp., 3) s not placed on or adjacent to vertces of C. Then we assgn color 2 to both v and y (resp., color 3 to y) foracolorng of H that s correct on P and agrees on the boundary wth that of C n G 1.

8 870 M.O. Albertson, J.P. Hutchnson / European Journal of Combnatorcs 25 (2004) Case (). Suppose c 1 (x) = 3. We transform the plane graph H by addng an edge jonng u and x n the outer face of H.Ifc 1 (y) = 2 (resp., 4), we 4-color H wth {1, 3, 4, 5} (resp., {1, 2, 3, 5})makngsureu and x (resp., u,v,andx)arecorrectly colored. Then as n the precedng paragraph we correct the colorng on P and assgn color 2 to v and y (resp., color 4 to y). In ths way the 5-precolorng of P always extends to G. The corollary below s a specfc verson of Theorem 4. Itsalsoastrengthenng of the orgnal result n the sense that the lower bound on the wdth s much smaller. Corollary 7.1. If G s embedded n a surface S of Euler genus g > 0 and w(g) 64(2 g 1),thenχ(G) 5. Proof. Follow the proof of Theorem 7 assumng that P =. Corollary 7.2. Suppose G s embedded n S, a surface of Euler genus g > 0 such that w(g) 64(2 g 1). Ifd(P) 4, thenany (5 + q)-colorng of G[P] n whch each component s q-colored extends to a (5 + q)-colorng of G. Proof. Corollary 7.1 mples that G Theorem 2. s 5-colorable and the result follows from 4. When a subgraph s precolored We close wth a 5-colorng extenson theorem n whch each precolored component s 2-colored. Theorem 8 s a specfc realzaton of Theorem 6. Theorem 8. Suppose G s embedded n S a surface of Euler genus g > 0. If,for 1 kdameter (G[P ]) D, d(p) 18, and w P (G) (16D + 408)(2 g 1), then any 5-colorng of G[P] n whcheach component s 2-colored extends to a 5-colorng of G. Proof. We omt detals that are dentcal to those n the proof of Theorem 7. Weassume that G s atrangulaton of S.Weknow that G contans a planarzng collecton of chordless cycles C 1, C 2,...,C s where s g such that for 1 j s, dst(c, C j ) 2d 2D + 50 [13, 15]. We let C 1, C 2,...,C s denote the cycles that detour around the components of G[P]. Specfcally C j s homotopc to C j, dst(c j, P) 9, and dst(c, C j ) 2D (D + 18) 14. We create the graph G 1 by makng the subgraph G C,4 orderly for each s.thenfor 1 s we apply Lemma 1 wth C = C 1.WeletC denote the resultng nce cycle, homotopc to C and lyng wthn G C,4. Thsgvesthefnal planarzng set of cycles {C 1,...,C s }.Wehavedst(C, C j ) 2d 36 8 = 6anddst(P, C ) 5. Next we form a plane trangulaton n two stages. Frst we cut out the planarzng cycles and replace each wth ether one or two vertces. Ther dstance from P s at least 5 and ther dstance from each other s at least 6. Next we look at P. Sncew P (G) >0, each component of P s contractble. For 1 k remove edges from G[P ] untl the resultng component s a tree. Contract the th component to obtan the vertex v and delete multple edges. The resultng graph G 0 s atrangulaton of the plane.

9 M.O. Albertson, J.P. Hutchnson / European Journal of Combnatorcs 25 (2004) Suppose c s a 4-colorng of G 0 usng colors {1, 2, 3, 4}.Asbeforewearrange that for 1 j s, fc s a 2-sded cycle, then c(x ) = c(y ).Wethencolor the vertces n G 1 G[P] by transferrng the colorng of G 0 and colorng each of the nce planarzng cycles. We stll need to fx the colorng n G[P]. For1 k we 2-colorthevertces of G[P ] usng the colors c(v ) and 5. The color classes n ths 2-colorng are dentcal to the color classes n the hypotheszed precolorng of G[P ]. Suppose for one partcular, c(v ) = 1andthatthe two color classes n G[P ] are assgned the colors 2 and 3 n the precolorng. Frst perform a (3, 5)-Kempe change at every vertex n G[P ] that s colored 5. Now every vertex n G[P ] that s supposed to be colored 3 s. Second perform a (1, 5)-Kempe change at every vertex n G[P ] that s colored 1. Fnally, perform a (2, 5)-Kempe change at every vertex n G[P ] that s currently colored 5. These three Kempe changes have the effect of makng the colors on G[P ] agree wth those of the precolorng. Furthermore these color changes are confned to P N(P ) N 2 (P ) N 3 (P ).Thearguments for other possbleprecolorngs are ether smlar or smpler [3]. The last step s to color the vertces of G that were deleted to create G 1.Allofthese vertces are nteror to contractble 3- or 4-cycles that are wthn dstance 4 of some C. We wll color these ndependently n exactly the same manner as was done n the proof of Theorem 7. If one wanted an extenson theorem n whch a 6-colorng of G[P] wth each component 2-colored extends to a 6-colorng of all of G, thend(p) 18 s needed. The requred hypothess on the relatve wdth s that w P (G) 208(2 g 1).Ths latter hypothess can be replaced by assumng both that dameter (G[P ]) D and w(g) (D+1)208(2g 1). In ether case the proofs are easer than the proof of Theorem 8. References [1] M.O. Albertson, You can t pant yourself nto a corner, J. Combn. Theory Ser. B 78 (1998) [2] M.O. Albertson, J.P. Hutchnson, Extendng colorngs of locally planar graphs, J. Graph Theory 36 (2001) [3] M.O. Albertson, J.P. Hutchnson, Graph color extensons: when Hadwger s conjecture and embeddngs help, Electron. J. Combn. 9 (2002 ) R37 (10 pages). [4] M.O. Albertson, J.P. Hutchnson, The ndependence rato and genus of a graph, Trans. Amer. Math. Soc. 226 (1977) [5] M.O. Albertson, W. Stromqust, Locally planar torodal graphs are 5-colorable, Proc. Amer. Math. Soc. 84 (1982) [6] K. Appel, W. Haken, Every planar map s four-colorable, Bull. Amer. Math. Soc. 82 (1976) [7] J.P. Hutchnson, A fve-color theorem for graphs on surfaces, Proc. Amer. Math. Soc. 90 (1984) [8] B. Mohar, C. Thomassen, Graphs on Surfaces, Johns Hopkns Unv. Press, Baltmore, [9] N. Robertson, D. Sanders, P. Seymour, R. Thomas, The four colour theorem, J. Combn. Theory Ser. B 70 (1997) [10] W. Stromqust, Manuscrpt, [11] C. Thomassen, Color-crtcal graphs on a fxed surface, 1996 (preprnt). [12] C. Thomassen, Color-crtcal graphs on a fxed surface, J. Combn. Theory Ser. B 70 (1997) [13] C. Thomassen, Fve-colorng maps on surfaces, J. Combn. Theory Ser. B 59 (1993) [14] Z. Tuza, Graph colorngs wth local constrants a survey, Dscuss. Math. Graph Theory 17 (1997) [15] X. Yu, Dsjont path, planarzng cycles, and k-walks, Trans. Amer. Math. Soc. 349 (4) (1997)