A BANDWIDTH THEOREM FOR APPROXIMATE DECOMPOSITIONS

Transcription

1 A BANDWIDTH THEOREM FOR APPROXIMATE DECOMPOSITIONS PADRAIG CONDON, JAEHOON KIM, DANIELA KÜHN AND DERYK OSTHUS Abstact. We povde a degee condton on a egula n-vetex gaph G whch ensues the exstence of a nea optmal pacng of any famly H of bounded degee n-vetex -chomatc sepaable gaphs nto G. In geneal, ths degee condton s best possble. Hee a gaph s sepaable f t has a sublnea sepaato whose emoval esults n a set of components of sublnea sze. Equvalently, the sepaablty condton can be eplaced by that of havng small bandwdth. Thus ou esult can be vewed as a veson of the bandwdth theoem of Böttche, Schacht and Taaz n the settng of appoxmate decompostons. Moe pecsely, let δ be the nfmum ove all δ 1/2 ensung an appoxmate K -decomposton of any suffcently lage egula n-vetex gaph G of degee at least δn. Now suppose that G s an n-vetex gaph whch s close to -egula fo some (δ + o(1))n and suppose that H 1,..., H t s a sequence of bounded degee n-vetex -chomatc sepaable gaphs wth e(h) (1 o(1))e(g). We show that thee s an edge-dsont pacng of H1,..., Ht nto G. If the H ae bpatte, then (1/2 + o(1))n s suffcent. In patcula, ths yelds an appoxmate veson of the tee pacng conectue n the settng of egula host gaphs G of hgh degee. Smlaly, ou esult mples appoxmate vesons of the Obewolfach poblem, the Alspach poblem and the exstence of esolvable desgns n the settng of egula host gaphs of hgh degee. 1. Intoducton Statng wth Dac s theoem on Hamlton cycles, a successful eseach decton n extemal combnatocs has been to fnd appopate mnmum degee condtons on a gaph G whch guaantee the exstence of a copy of a (possbly spannng) gaph H as a subgaph. On the othe hand, seveal mpotant questons and esults n desgn theoy as fo the exstence of a decomposton of K n nto edge-dsont copes of a (possbly spannng) gaph H, o moe geneally nto a sutable famly of gaphs H 1,..., H t. Hee, we combne the two dectons: athe than fndng ust a sngle spannng gaph H n a dense gaph G, we see (appoxmate) decompostons of a dense egula gaph G nto edge-dsont copes of spannng spase gaphs H. A specfc nstance of ths s the ecent poof of the Hamlton decomposton conectue and the 1-factozaton conectue fo lage n [12]: the fome states that fo n/2, evey -egula n-vetex gaph G has a decomposton nto Hamlton cycles and at most one pefect matchng, the latte povdes the coespondng theshold fo decompostons nto pefect matchngs. In ths pape, we estct ouselves to appoxmate decompostons, but acheve asymptotcally best possble esults fo a much wde class of gaphs than matchngs and Hamlton cycles Pevous esults: degee condtons fo spannng subgaphs. Mnmum degee condtons fo spannng subgaphs have been obtaned manly fo (Hamlton) cycles, tees, factos and bounded degee gaphs. We now befly dscuss seveal of these. Recall that Dac s theoem states that any n-vetex gaph G wth mnmum degee at least n/2 contans a Hamlton cycle. Moe geneally, Abbas s poof [1] of the El-Zaha conectue detemnes the mnmum degee theshold fo the exstence of a copy of H n G whee H s a spannng unon of vetex-dsont cycles (the theshold tuns out to be (n + odd H )/2, whee odd H denotes the numbe of odd cycles n H). Date: Octobe 4, The eseach leadng to these esults was patally suppoted by the EPSRC, gant no. EP/N019504/1 (D. Kühn), and by the Royal Socety and the Wolfson Foundaton (D. Kühn). The eseach was also patally suppoted by the Euopean Reseach Councl unde the Euopean Unon s Seventh Famewo Pogamme (FP/ ) / ERC Gant (J. Km and D. Osthus). 1

2 2 PADRAIG CONDON, JAEHOON KIM, DANIELA KÜHN AND DERYK OSTHUS Komlós, Sáözy and Szemeéd [33] poved a conectue of Bollobás by showng that a mnmum degee degee of n/2 + o(n) guaantees evey bounded degee n-vetex tee as a subgaph (ths was late stengthened n [35, 13, 26]). An F -facto n a gaph G s a set of vetex-dsont copes of F coveng all vetces of G. The Hanal-Szemeéd theoem [24] mples that the mnmum degee theshold fo the exstence of a K -facto s (1 1/)n. Ths was genealsed to th powes of Hamlton cycles by Komlós, Sáözy and Szemeéd [34]. The theshold fo abtay F -factos was detemned by Kühn and Osthus [38], and s gven by (1 c(f ))n+o(1), whee c(f ) satsfes 1/χ(F ) c(f ) 1/(χ(F ) 1) and can be detemned explctly (e.g. c(c 5 ) = 2/5, n accodance wth Abbas s esult). A fa-eachng genealsaton of the Hanal-Szemeéd theoem [24] would be povded by the Bollobás-Catln-Elddge (BEC) conectue. Ths would mply that evey n-vetex gaph G of mnmum degee at least (1 1/( + 1))n contans evey n-vetex gaph H of maxmum degee at most as a subgaph. Patal esults nclude the poof fo = 3 and lage n by Csaba, Shooufandeh and Szemeéd [14] and bounds fo lage by Kaul, Kostocha and Yu [28]. Bollobás and Komlós conectued that one can mpove on the BEC-conectue fo gaphs H wth a lnea stuctue: any n-vetex gaph G wth mnmum degee at least (1 1/ + o(1))n contans a copy of evey n-vetex -chomatc gaph H wth bounded maxmum degee and small bandwdth. Hee an n-vetex gaph H has bandwdth b f thee exsts an odeng v 1,..., v n of V (H) such that all edges v v E(H) satsfy b. Thoughout the pape, by H beng -chomatc we mean χ(h). Ths conectue was esolved by the bandwdth theoem of Böttche, Schacht and Taaz [9]. Note that whle ths esult s essentally best possble when consdeng the class of -chomatc gaphs as a whole (consde e.g. K -factos), the esults n [1, 38] mentoned above show that thee ae many gaphs H fo whch the actual theshold s sgnfcantly smalle (e.g. the C 5 -factos mentoned above). The noton of bandwdth s elated to the concept of sepaablty: An n-vetex gaph H s sad to be η-sepaable f thee exsts a set S of at most ηn vetces such that evey component of H \ S has sze at most ηn. We call such a set an η-sepaato of H. In geneal, the noton of havng small bandwdth s moe estctve than that of beng sepaable. Howeve, fo gaphs wth bounded maxmum degee, t tuns out that these notons ae actually equvalent (see [8]) Pevous esults: (appoxmate) decompostons nto lage gaphs. We say that a collecton H = {H 1,..., H s } of gaphs pacs nto G f thee exst pawse edge-dsont copes of H 1,..., H s n G. In cases whee H conssts of copes of a sngle gaph H we efe to ths pacng as an H-pacng n G. If H pacs nto G and e(h) = e(g) (whee e(h) = H H e(h)), then we say that G has a decomposton nto H. Once agan, f H conssts of copes of a sngle gaph H, we efe to ths as an H-decomposton of G. Infomally, we efe to a pacng whch coves almost all edges of the host gaph G as an appoxmate decomposton. As n the pevous secton, most attenton so fa has focussed on (Hamlton) cycles, tees, factos, and gaphs of bounded degee. Indeed, a classcal constucton of Walec gong bac to the 19th centuy guaantees a decomposton of K n nto Hamlton cycles wheneve n s odd. As mentoned eale, ths was extended to Hamlton decompostons of egula gaphs G of hgh degee by Csaba, Kühn, Lo, Osthus and Teglown [12] (based on the exstence of Hamlton decompostons n obustly expandng gaphs poved n [37]). A dffeent genealsaton of Walec s constucton s gven by the Alspach poblem, whch ass fo a decomposton of K n nto cycles of gven length. Ths was ecently esolved by Byant, Hosley and Petteson [10]. A futhe famous open poblem n the aea s the tee pacng conectue of Gyáfás and Lehel, whch says that fo any collecton T = {T 1,..., T n } of tees wth V (T ) =, the complete gaph K n has a decomposton nto T. Ths was ecently poved by Joos, Km, Kühn and Osthus [27] fo the case whee n s lage and each T has bounded degee. The cucal tool fo ths was the blow-up lemma fo appoxmate decompostons of ε-egula gaphs G by Km, Kühn, Osthus and Tyomyn [30]. In patcula, ths lemma mples that f H s a famly of bounded degee n-vetex gaphs wth e(h) (1 o(1)) ( n 2), then Kn has an appoxmate decomposton nto H. Ths genealses eale esults of Böttche, Hladỳ, Pguet and Taaz [7] on tee pacngs, as well as esults of Messut, Rödl and Schacht [39] and Febe, Lee and Mousset [17] on pacng

3 A BANDWIDTH THEOREM FOR APPROXIMATE DECOMPOSITIONS 3 sepaable gaphs. Vey ecently, Allen, Böttche, Hladỳ and Pguet [2] wee able to show that one can n fact fnd an appoxmate decomposton of K n nto H povded that the gaphs n H have bounded degeneacy and maxmum degee o(n/ log n). Ths mples an appoxmate veson of the tee pacng conectue when the tees have maxmum degee o(n/ log n). The latte mpoves a bound of Febe and Samot [18] whch follows fom the wo on pacng (spannng) tees n andom gaphs. An mpotant type of decomposton of K n s gven by esolvable desgns: a esolvable F - desgn conssts of a decomposton nto F -factos. Ray-Chaudhu and Wlson [42] poved the exstence of esolvable K -desgns n K n (subect to the necessay dvsblty condtons beng satsfed). Ths was genealsed to abtay F -desgns by Dues and Lng [16] Man esult: pacng sepaable gaphs of bounded degee. Ou man esult povdes a degee condton whch ensues that G has an appoxmate decomposton nto H fo any collecton H of -chomatc η-sepaable gaphs of bounded degee. As dscussed below, ou degee condton s best possble n geneal (unless one has addtonal nfomaton about the gaphs n H). By the ema at the end of Secton 1.1 eale, one can eplace the condton of beng η-sepaable by that of havng bandwdth at most ηn n Theoem 1.2. Thus ou esult mples a veson of the bandwdth theoem of [9] n the settng of appoxmate decompostons. To state ou esult, we fst ntoduce the appoxmate K -decomposton theshold δ eg fo egula gaphs. Defnton 1.1 (Appoxmate K -decomposton theshold fo egula gaphs). Fo each N\{1}, let δ eg be the nfmum ove all δ 0 satsfyng the followng: fo any ε > 0, thee exsts n 0 N such that fo all n n 0 and δn evey n-vetex -egula gaph G has a K -pacng consstng of at least (1 ε)e(g)/e(k ) copes of K. Roughly speang, we wll pac -chomatc gaphs H nto egula host gaphs G of degee at least δ eg n. Actually t tuns out that t suffces to assume that H s almost -chomatc n the sense that H has a ( + 1)-coloung whee one colou s used only aely. Moe pecsely, we say that H s (, η)-chomatc f thee exsts a pope coloung of the gaph H obtaned fom H by deletng all ts solated vetces wth + 1 colous such that one of the colou classes has sze at most η V (H ). A smla featue s also pesent n [9]. Theoem 1.2. Fo all, N\{1}, 0 < ν < 1 and max{1/2, δ eg } < δ 1, thee exst ξ, η > 0 and n 0 N such that fo all n n 0 the followng holds. Suppose that H s a collecton of n-vetex (, η)-chomatc η-sepaable gaphs and G s an n-vetex gaph such that () (δ ξ)n δ(g) (G) (δ + ξ)n, () (H) fo all H H, () e(h) (1 ν)e(g). Then H pacs nto G. Note that ou esult holds fo any mno-closed famly H of -chomatc bounded degee gaphs by the sepaato theoem of Alon, Seymou and Thomas [3]. Moeove, note that snce H may consst e.g. of Hamlton cycles, the condton that G s close to egula s clealy necessay. Also, the condton max{1/2, δ eg } < δ s necessay. To see ths, f δeg 1/2 (whch holds f = 2), then we consde K n/2 1,n/2+1 whch does not even contan a sngle pefect matchng, let alone an appoxmate decomposton nto pefect matchngs. If δ eg > 1/2 (whch holds f 3), then fo any δ < δ eg, the defnton of δeg ensues that thee exst abtaly lage egula gaphs G of degee at least δn wthout an appoxmate decomposton nto copes of K. As a dsont unon of a sngle copy of K wth n solated vetces satsfes (), ths shows that the condton of max{1/2, δ eg } < δ s shap when consdeng the class of all -chomatc sepaable gaphs (though as n the case of embeddng a sngle copy of some H nto G, t may be possble to mpove the degee bound fo cetan famles H). To obtan explct estmates fo δ eg, we also ntoduce the appoxmate K -decomposton theshold δ 0+ fo gaphs of lage mnmum degee.

4 4 PADRAIG CONDON, JAEHOON KIM, DANIELA KÜHN AND DERYK OSTHUS Defnton 1.3 (Appoxmate K -decomposton theshold). Fo each N\{1}, let δ 0+ be the nfmum ove all δ 0 satsfyng the followng: fo any ε > 0, thee exsts n 0 N such that any n-vetex gaph G wth n n 0 and δ(g) δn has a K -pacng consstng of at least (1 ε)e(g)/e(k ) copes of K. It s easy to see that δ eg 2 = δ2 0+ = 0 and δ eg δ 0+. The value of δ0+ has been subect to much attenton ecently: one eason s that by esults of [5, 19], fo 3 the appoxmate decomposton theshold δ 0+ s equal to the analogous theshold δ dec whch ensues a full K - decomposton of any n-vetex gaph G wth δ(g) (δ dec + o(1))n whch satsfes the necessay dvsblty condtons. A beautful conectue (due to Nash-Wllams n the tangle case and Gustavsson n the geneal case) would mply that δ dec = 1 1/( + 1) fo 3. On the othe hand fo 3, t s easy to modfy a well-nown constucton (see Poposton 3.7) to show that δ eg 1 1/( + 1). Thus the conectue would mply that δ eg = δ 0+ = δ dec = 1 1/( + 1) fo 3. A esult of Doss [15] mples that δ3 0+ 9/10, and a vey ecent esult of Montgomey [40] mples that δ /(100) (see Lemma 3.10). Wth these bounds, the followng coollay s mmedate. Coollay 1.4. Fo all, N\{1} and 0 < ν, δ < 1, thee exst ξ > 0 and n 0 N such that fo n n 0 the followng holds fo evey n-vetex gaph G wth (δ ξ)n δ(g) (G) (δ + ξ)n. () Let T be a collecton of tees such that fo all T T we have T n and (T ). Futhe suppose δ > 1/2 and e(t ) (1 ν)e(g). Then T pacs nto G. () Let F be an n-vetex gaph consstng of a unon of vetex-dsont cycles and let F be a collecton of copes of F. Futhe suppose δ > 9/10 and e(f) (1 ν)e(g). Then F pacs nto G. () Let C be a collecton of cycles, each on at most n vetces. Futhe suppose δ > 9/10 and e(c) (1 ν)e(g). Then C pacs nto G. (v) Let n be dvsble by and let K be a collecton of n-vetex K -factos. Futhe suppose δ > 1 1/(100) and e(k) (1 ν)e(g). Then K pacs nto G. Note that () can be vewed as an appoxmate veson of the tee pacng conectue n the settng of dense (almost) egula gaphs. In a smla sense, () elates to the Obewolfach conectue, () elates to the Alspach poblem and (v) elates to the exstence of esolvable desgns n gaphs. Moeove, the featue that Theoem 1.2 allows us to effcently pac (, η)-chomatc gaphs (athe than -chomatc gaphs) gves seveal addtonal consequences, fo example: f the cycles of F n () ae all suffcently long, then we can eplace the condton δ > 9/10 by δ > 1/2. If we dop the assumpton of beng G close to egula, then one can stll as fo the sze of the lagest pacng of bounded degee sepaable gaphs. Fo example, t was shown n [12] that evey suffcently lage gaph G wth δ(g) n/2 contans at least (n 2)/8 edge-dsont Hamlton cycles. The followng esult gves an appoxmate answe to the above queston n the case when H conssts of (almost) bpatte gaphs. Theoem 1.5. Fo all N, 1/2 < δ 1 and ν > 0, thee exst η > 0 and n 0 N such that fo all n n 0 the followng holds. Suppose that H s a collecton of n-vetex (2, η)-chomatc η-sepaable gaphs and G s an n-vetex gaph such that () δ(g) δn, () (H) fo all H H, () e(h) (δ+ 2δ 1 ν)n 2 4. Then H pacs nto G. The esult n geneal cannot be mpoved: Indeed, fo δ > 1/2 the numbe of edges of the densest egula spannng subgaph of G s close to (δ + 2δ 1)n 2 /4 (see [11]). So the bound n () s asymptotcally optmal e.g. f n s even and H conssts of Hamlton cycles. We dscuss

5 A BANDWIDTH THEOREM FOR APPROXIMATE DECOMPOSITIONS 5 the vey mno modfcatons to the poof of Theoem 1.2 whch gve Theoem 1.5 at the end of Secton 6. We ase the followng open questons: We conectue that the eo tem νe(g) n condton () of Theoem 1.2 can be mpoved. Note that t cannot be completely emoved unless one assumes some dvsblty condtons on G. Howeve, even addtonal dvsblty condtons wll not always ensue a full decomposton unde the cuent degee condtons: ndeed, fo C 4, the mnmum degee theshold whch guaantees a C 4 -decomposton of a gaph G s close to 2n/3, and the extemal example s close to egula (see [5] fo detals, moe geneally, the decomposton theshold of an abtay bpatte gaph s detemned n [19]). It would be nteestng to now whethe the condton on sepaablty can be omtted. Note howeve, that f we do not assume sepaablty, then the degee condton may need to be stengthened. It would be nteestng to now whethe one can elax the maxmum degee condton n assumpton () of Theoem 1.2, e.g. fo the class of tees. Gven the ecent pogess on the exstence of decompostons and desgns n the hypegaph settng and the coespondng mnmum degee thesholds [29, 20, 21], t would be nteestng to genealse (some of) the above esults to hypegaphs. Ou man tool n the poof of Theoem 1.2 wll be the ecent blow-up lemma fo appoxmate decompostons by Km, Kühn, Osthus and Tyomyn [30]: oughly speang, gven a set H of n-vetex bounded degee gaphs and an n-vetex gaph G wth e(h) (1 o(1))e(g) consstng of supe-egula pas, t guaantees a pacng of H n G (such supe-egula pas ase fom applcatons of Szemeéd s egulaty lemma). Theoem 3.15 gves the pecse statement of the specal case that we shall apply (note that the ognal blow-up lemma of Komlós, Sáözy and Szemeéd [31] coesponds to the case whee H conssts of a sngle gaph). Subsequently, Theoem 1.2 has been used as a ey tool n the esoluton of the Obewolfach poblem n [22]. Ths was posed by Rngel n 1967, gven an n-vetex gaph H consstng of vetex-dsont cycles, t ass fo a decomposton of K n nto copes of H (f n s odd). In fact, the esults n [22] go consdeably beyond the settng of the Obewolfach poblem, and mply e.g. a postve esoluton also to the Hamlton-Wateloo poblem. 2. Outlne of the agument Consde a gven collecton H of -chomatc η-sepaable gaphs wth bounded degee and a gven almost-egula gaph G as n Theoem 1.2. We wsh to pac H nto G. The appoach wll be to decompose G nto a bounded numbe of hghly stuctued subgaphs G t and patton H nto a bounded numbe of collectons H t. We then am to pac each H t nto G t. As descbed below, fo each H H t, most of the edges wll be embedded va the blow-up lemma fo appoxmate decompostons poved n [30]. As a pelmnay step, we fst apply Szemeéd s egulaty lemma (Lemma 3.5) to G to obtan a educed multgaph R whch s almost egula. Hee each edge e of R coesponds to a bpatte ε-egula subgaph of G and the densty of these subgaphs does not depend on e. We can then apply a esult of Pppenge and Spence on the chomatc ndex of egula hypegaphs and the defnton of δ eg to fnd an appoxmate decomposton of the educed multgaph R nto almost K -factos. Moe pecsely, we fnd a set of edge-dsont copes of almost K -factos coveng almost all edges of R, whee an almost K -facto s a set of vetex-dsont copes of K coveng almost all vetces of R. Ths appoxmate decomposton tanslates nto the exstence of an appoxmate decomposton of G nto (almost-)k -facto blow-ups. Hee a K -facto blow-up conssts of a bounded numbe of clustes V 1,..., V whee each pa (V, V ) wth ( 1)/ = ( 1)/ s ε-egula of densty d, and cucally d does not depend on,. We wsh to use the blow-up lemma fo appoxmate decompostons (Theoem 3.15) to pac gaphs nto each K -facto blow-up. Ideally, we would le to splt H nto a bounded numbe of subcollectons H t,s and pac each H t,s nto a sepaate K -facto blow-up G t,s, whee the G t,s G ae all edge-dsont.

6 6 PADRAIG CONDON, JAEHOON KIM, DANIELA KÜHN AND DERYK OSTHUS Thee ae seveal obstacles to ths appoach. The fst obstacle s that () the K -facto blowups G t,s ae not spannng. In patcula, they do not contan the vetces n the exceptonal set V 0 poduced by the egulaty lemma. On the othe hand, f we am to embed an n-vetex gaph H H nto G, we must embed some vetces of H nto V 0. Howeve, Theoem 3.15 does not poduce an embeddng nto vetces outsde the K -facto blow-up. The second obstacle s that () the K -facto blow-ups ae not connected, wheeas H may cetanly be (hghly) connected. Ths s one sgnfcant dffeence to [9], whee the exstence of a stuctue smla to a blown-up powe of a Hamlton path n R could be utlsed fo the embeddng. A thd ssue s that () any esoluton of () and () needs to esult n a balanced pacng of the H H,.e. the condton e(h) (1 ν)e(g) means that fo most x V (G) almost all the ncdent edges need to be coveed. To ovecome the fst ssue, we use the fact that H s η-sepaable to choose a small sepaatng set S fo H and consde the small components of H S. To be able to embed (most of) H nto the K -facto blow-up, we need to add futhe edges to each K -facto blow-up so that the esultng augmented K -facto blow-ups have stong connectvty popetes. Fo ths, we patton V (G)\V 0 nto T dsont esevos Res 1,..., Res T, whee 1/T 1. We wll late embed some vetces of H nto V 0 usng the edges between Res t and V 0 (see Lemma 4.1). Hee we have to embed a vetex of H onto v V 0 usng only edges between v and Res t because we do not have any contol on the edges between v and a egulaty cluste V. We explan the eason fo choosng a patton nto many esevo sets (athe than choosng a sngle small esevo) below. We also decompose most of G nto gaphs G t,s so that each G t,s has vetex set V (G)\(Res t V 0 ) and s a K -facto blow-up. We then fnd spase bpatte gaphs F t,s G connectng Res t wth G t,s, bpatte gaphs F t G connectng Res t wth V 0 as well as spase gaphs G t G whch povde connectvty wthn Res t as well as between Res t and G t,s. The fact that G t,s and G t,s shae the same esevo fo s s pemts us to choose the esevo Res t to be sgnfcantly lage than V 0. Moeove, as Res t coves all vetces n V \V 0, f the gaphs F t ae appopately chosen, then almost all edges ncdent to the vetces n V 0 ae avalable to be used at some stage of the pacng pocess. Ou am s to pac each H t,s nto the augmented K -facto blow-up G t,s F t,s F t G t. To ensue that the esultng pacngs can be combned nto a pacng of all of the gaphs n H, we wll use the fact that the gaphs G t := s (G t,s F t,s ) F t G t efeed to n the fst paagaph ae edge-dsont fo dffeent t. We now dscuss how to fnd ths pacng of H t,s. Consde some H H t,s. We fst use the fact that H s sepaable to fnd a patton of H whch eflects the stuctue of (the augmentaton of) G t,s (see Secton 4). Then we constuct an appopate embeddng φ of pats of each gaph H H t,s nto Res t V 0 whch coves all vetces n Res t V 0 (ths maes cucal use of the fact that Res t s much lage than V 0 ). Late we am to use the blow-up lemma fo appoxmate decompostons (Theoem 3.15) to fnd an embeddng φ of the emanng vetces of H nto V (G)\(Res t V 0 ). When we apply Theoem 3.15, we use ts addtonal featues: n patcula, the ablty to pescbe appopate taget sets fo some of the vetces of H, to guaantee the consstency between the two embeddngs φ and φ. An mpotant advantage of the esevo patton whch helps us to ovecome obstacle () s the followng: the blow-up lemma fo appoxmate decompostons can acheve a nea optmal pacng,.e. t uses up almost all avalable edges. Ths s fa fom beng the case fo the pat of the embeddngs that use F t,s, F t and G t to embed vetces nto Res t V 0, whee the edge usage mght be compaatvely mbalanced and neffcent. (In fact, we wll ty to avod usng these edges as much as possble n ode to peseve the connectvty popetes of these gaphs. We wll use pobablstc allocatons to avod ove-usng any pats of F t,s, F t and G t.) Howeve, snce evey vetex n V (G 0 )\V 0 s a esevo vetex fo only a small popoton of the embeddngs, the esultng effect of these mbalances on the oveall leftove degee of the vetces n V (G 0 )\V 0 s neglgble. Fo V 0, we wll be able to assgn only low degee vetces of each H to ensue that thee wll always be edges of F t avalable to embed the ncdent edges (so the oveall leftove degee of the vetces n V 0 may be lage).

7 A BANDWIDTH THEOREM FOR APPROXIMATE DECOMPOSITIONS 7 The above dscusson motvates why we use many esevo sets whch cove all vetces n V (G)\V 0, athe than usng only one vetex set Res 1 fo all H H. Indeed, f some vetces of G only pefom the ole of esevo vetces, ths mght esult n an mbalance of the usage of edges ncdent to these vetces: some vetces n the esevo mght lose ncdent edges much faste o slowe than the vetces n the egulaty clustes. Apat fom the fact that a fast loss of the edges ncdent to one vetex can pevent us fom embeddng any futhe spannng gaphs nto G, a lage loss of the edges ncdent to the esevo s also poblematc n ts own ght. Indeed, snce we ae foced to use the edges ncdent to the esevo n ode to be able to embed some vetces onto vetces n V 0, ths would pevent us fom pacng any futhe gaphs. Anothe ssue s that the egulaty lemma only gves us ε-egula K -facto blow-ups whle we need supe-egula K -facto blow-ups n ode to use Theoem To ovecome ths ssue, we wll mae appopate adustments to each ε-egula K -facto blow-up. Ths means that the exceptonal set V 0 wll actually be dffeent fo each pa t, s of ndces. We can howeve use pobablstc aguments to ensue that ths does not sgnfcantly affect the oveall balance of the pacng. In patcula, fo smplcty, n the above poof setch we have gnoed ths ssue. The pape s ogansed as follows. We collect some basc tools n Secton 3, and we pove a lemma whch fnds a sutable patton of each gaph H H n Secton 4 (Lemma 4.1). We pove ou man lemma (Lemma 5.1) n Secton 5. Ths lemma guaantees that we can fnd a sutable pacng of an appopate collecton H t,s of -chomatc η-sepaable gaphs wth bounded degee nto a gaph consstng of a supe-egula K -facto blow-up G t,s and sutable connecton gaphs F t,s, F t and G t. In Secton 6, we wll patton G and H as descbed above. Then we wll epeatedly apply Lemma 5.1 to constuct a pacng of H nto G. 3. Pelmnaes 3.1. Notaton. We wte [t] := {1,..., t}. We often teat lage numbes as nteges wheneve ths does not affect the agument. The constants n the heaches used to state ou esults ae chosen fom ght to left. That s, f we clam that a esult holds fo 0 < 1/n a b 1, we mean thee exst non-deceasng functons f : (0, 1] (0, 1] and g : (0, 1] (0, 1] such that the esult holds fo all 0 a, b 1 and all n N wth a f(b) and 1/n g(a). We wll not calculate these functons explctly. We use the wod gaphs to efe to smple undected fnte gaphs, and efe to mult-gaphs as gaphs wth potentally paallel edges, but wthout loops. Mult-hypegaphs efe to (not necessaly unfom) hypegaphs wth potentally paallel edges. A -gaph s a -unfom hypegaph. A mult--gaph s a -unfom hypegaph wth potentally paallel edges. Fo a mult-hypegaph H and a non-empty set Q V (H), we defne mult H (Q) to be the numbe of paallel edges of H consstng of exactly the vetces n Q. We say that a mult-hypegaph has edge-multplcty at most t f mult H (Q) t fo all non-empty Q V (H). A matchng n a multhypegaph H s a collecton of pawse dsont edges of H. The an of a mult-hypegaph H s the sze of a lagest edge. We wte H G f two gaphs H and G ae somophc. Fo a collecton H of gaphs, we let v(h) := H H V (H). We say a patton V 1,..., V of a set V s an equpatton f V V 1 fo all, []. Fo a mult-hypegaph H and A, B V (H), we let E H (A, B) denote the set of edges n H ntesectng both A and B. We defne e H (A, B) := E H (A, B). Fo v V (H) and A V (H), we let d H,A (v) := {e E(H) : v e, e\{v} A}. Let d H (v) := d H,V (H) (v). Fo u, v V (H), we defne c H (u, v) := {e E(H) : {u, v} e}. Let (H) = max{d H (v) : v V (H)} and δ(h) := mn{d H (v) : v V (H)}. Fo a gaph G and sets X, A V (G), we defne N G,A (X) := {w A : uw E(G) fo all u X} and N G (X) := N G,V (G) (X). Thus N G (X) s the common neghbouhood of X n G and N G,A ( ) = A. Fo a set X V (G), we defne NG d (X) V (G) to be the set of all vetces of dstance at most d fom a vetex n X. In patcula, NG d (X) = fo d < 0. Note that N G(X) and NG 1 (X) ae dffeent n geneal as e.g. vetces wth a sngle edge to X ae ncluded n the latte. Moeove, note that N G (X) NG 1 (X). We say a set I V (G) n a gaph G s -ndependent f fo any two dstnct

8 8 PADRAIG CONDON, JAEHOON KIM, DANIELA KÜHN AND DERYK OSTHUS vetces u, v I, the dstance between u and v n G s at least (thus a 2-ndependent set I s an ndependent set). If A, B V (G) ae dsont, we wte G[A, B] fo the bpatte subgaph of G wth vetex classes A, B and edge set E G (A, B). Fo two functons φ : A B and φ : A B wth A A =, we let φ φ be the functon fom A A to B B such that fo each x A A, { (φ φ φ(x) f x A, )(x) := φ (x) f x A. Fo gaphs H and R wth V (R) [] and an odeed patton (X 1,..., X ) of V (H), we say that H admts the vetex patton (R, X 1,..., X ), f H[X ] s empty fo all [], and fo any, [] wth we have that e H (X, X ) > 0 mples E(R). We say that H s ntenally q-egula wth espect to (R, X 1,..., X ) f H admts (R, X 1,..., X ) and H[X, X ] s q-egula fo each E(R). We wll often use the followng Chenoff bound (see e.g. Theoem A.1.16 n [4]). Lemma 3.1. [4] Suppose X 1,..., X n ae ndependent andom vaables such that 0 X b fo all [n]. Let X := X X n. Then fo all t > 0, P[ X E[X] t] 2e t2 /(2b 2 n) Tools nvolvng ε-egulaty. In ths subsecton, we ntoduce the defntons of (ε, d)- egulaty and (ε, d)-supe-egulaty. We then state a sutable fom of the egulaty lemma fo ou pupose. We wll also state an embeddng lemma (Lemma 3.6) whch we wll use late to pove ou man lemma (Lemma 5.1). We say that a bpatte gaph G wth vetex patton (A, B) s (ε, d)-egula f fo all sets A A, B B wth A ε A, B ε B, we have e G(A,B ) A B d < ε. Moeove, we say that G s ε-egula f t s (ε, d)-egula fo some d. If G s (ε, d)-egula and d G (a) = (d ± ε) B fo a A and d G (b) = (d ± ε) A fo b B, then we say G s (ε, d)-supe-egula. We say that G s (ε, d) + -(supe)-egula f t s (ε, d )-(supe)-egula fo some d d. Fo a gaph R on vetex set [], and dsont vetex subsets V 1,..., V of V (G), we say that G s (ε, d) + -(supe)-egula wth espect to the vetex patton (R, V 1,..., V ) f G[V, V ] s (ε, d) + -(supe)-egula fo all E(R). Beng (ε, d)-(supe)-egula wth espect to the vetex patton (R, V 1,..., V ) s defned analogously. The followng obsevatons follow dectly fom the defntons. Poposton 3.2. Let 0 < ε δ d 1. Suppose G s an (ε, d)-egula bpatte gaph wth vetex patton (A, B) and let A A, B B wth A / A, B / B δ. Then G[A, B ] s (ε/δ, d)-egula. Poposton 3.3. Let 0 < ε δ d 1. Suppose G s an (ε, d)-egula bpatte gaph wth vetex patton (A, B). If G s a subgaph of G wth V (G ) = V (G) and e(g ) (1 δ)e(g), then G s (ε + δ 1/3, d)-egula. Poposton 3.4. Let 0 < ε d 1. Suppose G s an (ε, d)-egula bpatte gaph wth vetex patton (A, B). Let A := {a A : d G (a) (d ± ε) B } and B := {b B : d G (b) (d ± ε) B }. Then A 2ε A and B 2ε B. The next lemma s a degee veson of Szemeéd s egulaty lemma (see e.g. [36] on how to deve t fom the ognal veson). Lemma 3.5 (Szemeéd s egulaty lemma). Suppose M, M, n N wth 0 < 1/n 1/M ε, 1/M < 1 and d > 0. Then fo any n-vetex gaph G, thee exst a patton of V (G) nto V 0, V 1,..., V and a spannng subgaph G G satsfyng the followng. () M M, () V 0 εn, () V = V fo all, [], (v) d G (v) > d G (v) (d + ε)n fo all v V (G), (v) e(g [V ]) = 0 fo all [],

9 A BANDWIDTH THEOREM FOR APPROXIMATE DECOMPOSITIONS 9 (v) fo all, wth 1 <, the gaph G [V, V ] s ethe empty o (ε, d, )-egula fo some d, [d, 1]. The next lemma allows us to embed a small gaph H nto a gaph G whch s (ε, d) + -egula wth espect to a sutable vetex patton (R, V 1,..., V ). In ou poof of Lemma 5.1 late on, popetes (B1) 3.6 and (B2) 3.6 wll help us to pescbe appopate taget sets fo some of the vetces when we apply the blow-up lemma fo appoxmate decompostons (Theoem 3.15). Thee, H wll be pat of a lage gaph that s embedded n seveal stages. (B1) 3.6 ensues that the embeddng of H s compatble wth constants asng fom eale stages and (B2) 3.6 wll ensue the exstence of suffcently lage taget sets when embeddng vetces x n late stages (each edge of M coesponds to the neghbouhood of such a vetex x). Lemma 3.6. Suppose n, N wth 0 < 1/n ε α, β, d, 1/ 1. Suppose that G, H ae gaphs and M s a mult-hypegaph on V (H) wth edge-multplcty at most. Suppose V 1,..., V ae pawse dsont subsets of V (G) wth βn V n fo all [], and X 1,..., X s a patton of V (H) wth X εn fo all []. Let f : E(M) [] be a functon, and fo all [] and x X, let A x V. Let R be a gaph on []. Suppose that the followng hold. (A1) 3.6 G s (ε, d) + -egula wth espect to (R, V 1,..., V ), (A2) 3.6 H admts the vetex patton (R, X 1,..., X ), (A3) 3.6 (H), (M) and the an of M s at most, (A4) 3.6 fo all, [], f f(e) = and e X, then E(R), (A5) 3.6 fo all [] and x X, we have A x α V. Then thee exsts an embeddng φ of H nto G such that (B1) 3.6 fo each x V (H), we have φ(x) A x, (B2) 3.6 fo each e M, we have N G (φ(e)) V f(e) (d/2) V f(e). Note that (A4) 3.6 mples fo all e E(M) that e X f(e) =. Poof. Fo each x V (H), let e x := N H (x) and M be a mult-hypegaph on vetex set V (H) wth E(M ) = {e x : x V (H)}. Snce a vetex x V (H) belongs to e y only when y N H (x), we have d M (x) = d H (x). So M s a mult-hypegaph wth an at most and (M ). Let M := M M and fo each e E(M ), defne { Vf(e) f e E(M), B e := f e = e x E(M ) fo x V (H). Note that by (A3) 3.6, we have A x M has an at most, and (M ) (M) + (M ) 2. (3.1) Let V (H) := {x 1,..., x m }, and fo each [m], we let Z := {x 1,..., x }. We wll teatvely extend patal embeddngs φ 0,..., φ m of H nto G n such a way that the followng hold fo all m. (Φ1) 3.6 φ embeds H[Z ] nto G, (Φ2) 3.6 φ (x ) A x, fo all [], (Φ3) 3.6 fo all e M, we have N G (φ (e Z )) B e (d/2) e Z B e. Note that (Φ1) (Φ3)0 3.6 hold fo an empty embeddng φ 0 :. Assume that fo some [m], we have aleady defned an embeddng φ 1 satsfyng (Φ1) (Φ3) We wll constuct φ by choosng an appopate mage fo x. Let s [] be such that x X s, and let S := N G (φ 1 (Z e x )) B ex. Thus S V s. Snce Z 1 e x = Z e x, we have that (Φ3) mples Fo each e E(M ) contanng x, we consde S (d/2) Z e x αβn > (d/2) αβn > ε 1/3 n. (3.2) S e := N G (φ 1 (Z 1 e)) B e. By (Φ3) 1 3.6, we have S e (d/2) αβn > ε 1/3 n. (3.3)

10 10 PADRAIG CONDON, JAEHOON KIM, DANIELA KÜHN AND DERYK OSTHUS If e = N H (x) fo some x X s wth s [], then we have S e B e V s, and (A2) 3.6 mples that ss E(R). Moeove, note that f e M wth f(e) = s fo some s [], then S e B e = V s, and (A4) 3.6 mples that ss E(R). Thus n any case, (A1) 3.6 mples that G[V s, V s ] s (ε, d )-egula fo some d d. Hence, Poposton 3.2 wth (3.2) and (3.3) mples that G[S, S e ] s (ε 1/2, d )-egula. Let S e := {v S : d G,Se (v) < (d/2) S e }. By Poposton 3.4, we have S e 2ε 1/2 n. Thus S \ S e (3.1) S 2 2ε 1/2 n (3.2) 1. (3.4) e E(M ):x e We choose v S \ e E(M ):x e S e, and we extend φ 1 nto φ by lettng φ (x ) := v. Snce φ (x ) S = N G (φ 1 (Z e x )) B ex = N G (φ (Z N H (x ))) A x, (Φ1) 3.6 and (Φ2) 3.6 hold. Also, fo each e E(M ), f x / e, then as we have Z e = Z 1 e, N G (φ (Z e)) B e = N G (φ 1 (Z 1 e)) B e (Φ3) (d/2) Z e B e. If x e, then snce φ (x ) / S e and Z e = Z 1 e + 1, we have N G (φ (Z e)) B e N G (φ (x )) S e (d/2) S e (Φ3) (d/2) Z e B e. (3.5) Thus (Φ3) 3.6 holds. By epeatng ths untl we have embedded all vetces of H, we obtan an embeddng φ m satsfyng (Φ1) m 3.6 (Φ3)m 3.6. Let φ := φ m. Then (Φ2) m 3.6 mples that (B1) 3.6 holds, and (Φ3) m 3.6 togethe wth (A3) 3.6 and the defnton of B e mples that (B2) 3.6 holds Decomposton tools. In ths subsecton, we fst gve bounds on δ eg. The followng poposton povdes a lowe bound fo δ eg. The poof s only a slght extenson of the extemal constucton gven by Poposton 1.5 n [5], and thus we omt t hee. Poposton 3.7. Fo all N\{1, 2} we have δ eg 1 1/( + 1). It wll be convenent to use that fo 2 ths lowe bound mples max{1/2, δ eg } 1 1/. (3.6) Gven two gaphs F and G, let ( G F) denote the set of all copes of F n G. A functon ψ fom ( G F) to [0, 1] s a factonal F -pacng of G f F ( G ψ(f F):e F ) 1 fo each e E(G) (f we have equalty fo each e E(G) then ths s efeed to as a factonal F -decomposton). Let νf (G) be the maxmum value of F ( G F) ψ(f ) ove all factonal F -pacngs ψ of G. Thus νf (G) e(g)/e(f ) and νf (G) = e(g)/e(f ) f and only f G has a factonal F -decomposton. The followng vey ecent esult of Montgomey gves a degee condton whch ensues a factonal K -decomposton n a gaph. Theoem 3.8. [40] Suppose, n N and 0 < 1/n 1/ < 1. Then any n-vetex gaph G wth δ(g) (1 1/(100))n satsfes ν K (G) = e(g)/e(k ). The next esult due to Haxell and Rödl mples that a factonal K -decomposton gves se to the exstence of an appoxmate K -decomposton. Theoem 3.9. [25] Suppose n N wth 0 < 1/n ε < 1. Then any n-vetex gaph G has an F -pacng consstng of at least ν F (G) εn2 copes of F. Lemma Fo N\{1, 2}, we have δ eg δ 0+ and δ eg 3 δ / /(100). Moeove, δ eg 2 = δ 0+ 2 = 0 Poof. It s easy to see that Theoem 3.8 and Theoem 3.9 togethe mply that δ /(100). Moeove, Theoem 3.9 togethe wth a esult of Doss [5] mples that δ3 0+ 9/10. As any gaph can be decomposed nto copes of K 2, we have δ2 0+ = 0.

11 A BANDWIDTH THEOREM FOR APPROXIMATE DECOMPOSITIONS 11 In the emande of ths subsecton, we pove Lemma In the poof of Theoem 1.2, we wll apply t to obtan an appoxmate decomposton of the educed mult-gaph R nto almost K -factos (see Secton 6). We wll use the followng consequence of Tutte s -facto theoem. Theoem [11] Suppose n N and 0 < 1/n γ 1. If G s an n-vetex gaph wth δ(g) (1/2 + γ)n and (G) δ(g) + γ 2 n, then G contans a spannng -egula subgaph fo evey even wth δ(g) γn. The followng poweful esult of Pppenge and Spence [41] (based on the Rödl nbble) shows that evey almost egula mult--gaph wth small maxmum codegee has small chomatc ndex. Theoem [41] Suppose n, N and 0 < 1/n µ ε, 1/ < 1. Suppose H s an n-vetex mult--gaph satsfyng δ(h) (1 µ) (H), and c H (u, v) µ (H) fo all u v V (H). Then we can patton E(H) nto (1 + ε) (H) matchngs. We can now combne these tools to appoxmately decompose an almost egula mult-gaph G of suffcent degee nto almost K -factos. All vetces of G wll be used n almost all these factos except the vetces n a bad set V whch ae not used n any facto. Moeove, the factos come n T goups of equal sze such that paallel edges of G belong to dffeent goups. As explaned n Secton 2, we wll apply ths to the educed mult-gaph obtaned fom Szemeéd s egulaty lemma. Lemma Suppose n,, q, T N wth 0 < 1/n ε, σ, 1/T, 1/, 1/q, ν 1/2 and 0 < 1/n ξ ν < σ/2 < 1 and δ = max{1/2, δ eg } + σ and q dvdes T. Let G be an n-vetex mult-gaph wth edge-multplcty at most q, such that fo all v V (G) we have d G (v) = (δ ± ξ)qn. Then thee exsts a subset V V (G) wth V εn and dvdng V (G)\V, and thee exst qn pawse edge-dsont subgaphs F 1,1,..., F 1,κ, F 2,1,..., F T,κ wth κ = (δ ν ±ε) T ( 1) satsfyng the followng. (B1) 3.13 Fo each (t, ) [T ] [κ], we have that V (F t,) V (G)\V and F t, s a vetex-dsont unon of at least (1 ε)n/ copes of K, (B2) 3.13 fo each v V (G) \ V, we have {(t, ) [T ] [κ] : v V (F t,)} T κ εn, (B3) 3.13 fo all t [T ] and u, v V (G), we have { [κ] : u N Ft, (v)} 1. Poof. It suffces to pove the lemma fo the case when T = q. The geneal case then follows by elabellng. (We can splt each goup obtaned fom the T = q case nto T/q equal goups abtaly.) We choose a new constant µ such that 1/n µ ε, ξ, σ, 1/, 1/q. Fo an edge coloung φ : E(G) [q] and c [q], we let G c G be the subgaph wth edge set {e E(G) : φ(e) = c}. We wsh to show that thee exsts an edge-coloung φ : E(G) [q] satsfyng the followng fo all v V (G) and c [q]: (Φ1) 3.13 d G c(v) = (δ ± 2ξ)n, (Φ2) 3.13 G c s a smple gaph. Recall that e G (u, v) denotes the numbe of edges of G between u and v. Fo each {u, v} ( V (G) ) 2, we choose a set A{u,v} unfomly at andom fom ( [q] e G (u,v)). Fo each e E(G), we let φ(e) [q] be such that φ s bectve between E G (u, v) and A {u,v}. Ths ensues that (Φ2) 3.13 holds. It s easy to see that (Φ1) 3.13 also holds wth hgh pobablty by usng Lemma 3.1. Snce δ 1/2 + σ and ξ ν, σ, Theoem 3.11 mples that, fo each c [q], thee exsts a (δ ν)n-egula spannng subgaph G c of G c. (By adustng ν slghtly we may assume that (δ ν)n s an even ntege.) Snce δ ν > δ eg + σ/2 and 1/n µ, the gaph G c has a K -pacng Q c := {Q c 1,..., Qc t} of sze t := (δ ν µ)n2. (3.7) ( 1)

12 12 PADRAIG CONDON, JAEHOON KIM, DANIELA KÜHN AND DERYK OSTHUS Fo each c [q], let H c be the -gaph wth V (H c ) = V (G c ) and E(H c ) := {V (Q c ) : [t]}. By constucton of H c, we have (H c ) (Gc ) (δ ν)n 1 1. (3.8) As Q c s a K -pacng n G c, any pa {u, v} ( ) V (G) 2 belongs to at most one edge n H c. Thus fo {u, v} ( ) V (G) 2, Let V := c [q] c H c(u, v) 1. (3.9) { v V (G) : { [t] : v V (Q c )} < 1 } 1 (δ ν µ1/3 )n, and let V be a set consstng of the unon of V as well as at most 1 vetces abtaly chosen fom V (G)\V such that dvdes V (G)\V. Note that fo each c [q], we have e(g c ) e(q c ) 1 ( ) 2 (δ ν)n2 t (3.7) µn 2. 2 On the othe hand, snce G c s a (δ ν)n-egula gaph, we have V ( dg µ 1/3 c n (v) ( 1)d H c(v)) c [q] v V (G) = c [q] 2(e(G c ) e(q c )) µ 1/3 n 3qµn2 µ 1/3 n µ1/2 n. (3.10) Let H c be the -gaph wth V ( H c ) := V (G c ) \ V and E( H c ) := {e E(H c ) : e V = }. Note that fo any v V ( H c ) = V (H c ) \ V, d Hc(v) = d H c(v) ± u V c H c(u, v) (3.9) = d H c(v) ± V (3.10),(3.8) = (δ ν ± 2µ 1/3 )n. (3.11) 1 Note that we obtan the fnal equalty fom the defnton of V and the assumpton that v / V. Thus fo each c [q], we have δ( H c ) (1 µ 1/4 ) ( H c ). Togethe wth (3.9) and the fact that 1/n µ ε, 1/, 1/q, ths ensues that we can apply Theoem 3.12 to see that fo each c [q], E( H c ) can be pattoned nto κ := (δ ν+ε3 /q)n 1 matchngs M1 c,..., M κ c. Let M c := {M c : [κ ]} and M c := {M c : [κ ], M c < (1 ε)n/}. As M c n/ fo any [κ ] and c [q], we have Ths gves We let (δ ν 3µ 1/3 )n 2 ( 1) (3.10),(3.11) E( H c ) = [κ ] M c (ε3 /q + 3µ 1/3 )n 2 εn( 1) M c < Mc (1 ε)n + (κ M c )n. 2ε2 n q( 1). (3.12) κ := mn c [q] { Mc \ M c } = κ max c [q] { Mc } = (δ ν)n ± 2ε2 n/q. (3.13) 1 Thus, by pemutng ndces, we can assume that fo each c [q], we have M1 c,..., M κ c M c \M c. Fo each (c, ) [q] [κ], let F c, := Q c. :V (Q c ) M c

13 A BANDWIDTH THEOREM FOR APPROXIMATE DECOMPOSITIONS 13 The fact that M c \M c s a collecton of pawse edge-dsont matchngs of H c H c togethe wth (3.9) mples that, fo each c [q], the collecton {F c, : [κ]} conssts of pawse edgedsont subgaphs of G c G, each of whch s a unon of at least (1 ε)n/ vetex-dsont copes of K. Ths wth (Φ2) 3.13 shows that (B3) 3.13 holds. As G 1,..., G q ae pawse edge-dsont subgaphs, {F c, : (c, ) [q] [κ]} foms a collecton of pawse edge-dsont subgaphs of G. Thus (B1) 3.13 holds. Moeove, fo each c [q] and each vetex v V (G) \ V, we have { [κ] : v V (F c, )} {M {M c 1,..., M c κ} : v V (M)} {M M c : v V (M)} (κ κ) d Hc(v) κ + κ (3.11) κ εn/q. Thus (B2) 3.13 holds Gaph pacng tools. The followng two esults fom [30] wll allow us to pac many bounded degee gaphs nto appopate supe-egula blow-ups. Lemma 3.14 fst allows us to pac gaphs nto ntenally egula gaphs whch stll have bounded degee, and Theoem 3.15 allows us to pac the ntenally egula gaphs nto an appopate dense ε-egula gaph. The esults n [30] ae actually sgnfcantly moe geneal, manly because they allow fo moe geneal educed gaphs R. Lemma [30, Lemma 7.1] Suppose n,, q, s,, N wth 0 < 1/n ε 1/s 1/, 1/ and ε 1/q 1 and dvdes. Suppose that 0 < ξ < 1 s such that s 2/3 ξq. Let R be a gaph on [] consstng of / vetex-dsont copes of K. Let V 1,..., V be a patton of some vetex set V such that V = n fo all []. Suppose fo each [s], L s a gaph admttng the vetex patton (R, X 1,..., X ) such that (L ) and fo each E(R), we have s e(l [X, X ]) = (1 3ξ ± ξ)qn, =1 and X n. Also suppose that fo all [s] and [], we have sets W X such that W εn. Then thee exsts a gaph H on V whch s ntenally q-egula wth espect to (R, V 1,..., V ) and a functon φ whch pacs {L 1,..., L s } nto H such that φ(x ) V, and such that fo all dstnct, [s] and [], we have φ(w ) φ(w ) =. Theoem 3.15 (Blow-up lemma fo appoxmate decompostons [30, Theoem 6.1]). Suppose n, q, s,, N wth 0 < 1/n ε α, d, d 0, 1/q, 1/ 1 and 1/n 1/ and dvdes. Suppose that R s a gaph on [] consstng of / vetex-dsont copes of K. Suppose s d q (1 α/2)n and the followng hold. (A1) 3.15 G s (ε, d)-supe-egula wth espect to the vetex patton (R, V 1,..., V ). (A2) 3.15 H = {H 1,..., H s } s a collecton of gaphs, whee each H s ntenally q-egula wth espect to the vetex patton (R, X 1,..., X ), and X = V = n fo all []. (A3) 3.15 Fo all [s] and [], thee s a set W X wth W εn and fo each w W, thee s a set A w V wth A w d 0 n. (A4) 3.15 Λ s a gaph wth V (Λ) [s] =1 X and (Λ) (1 α)d 0 n such that fo all (, x) V (Λ) and [s], we have {x : (, x ) N Λ ((, x))} q 2. Moeove, fo all [s] and [], we have {(, x) V (Λ) : x X } ε X. Then thee s a functon φ pacng H nto G such that, wtng φ fo the estcton of φ to H, the followng hold fo all [s] and []. (B1) 3.15 φ (X ) = V, (B2) 3.15 φ (w) A w fo all w W, (B3) 3.15 fo all (, x)(, y) E(Λ), we have that φ (x) φ (y).

14 14 PADRAIG CONDON, JAEHOON KIM, DANIELA KÜHN AND DERYK OSTHUS 3.5. Mscellaneous. In the poof of Theoem 1.2, we often patton vaous gaphs nto pats wth cetan popetes. The next two lemmas wll allow us to obtan such pattons. Lemma 3.16 follows by consdeng a andom equpatton and applyng concentaton of the hypegeometc dstbuton. Lemma 3.17 can be poved by assgnng each edge of G to G 1,..., G s ndependently at andom accodng to (p 1,..., p s ), and applyng Lemma 3.1. We omt the detals. Lemma Suppose n, T, N wth 0 < 1/n 1/T, 1/ 1. Let G be an n-vetex gaph. Let V V (G) and let V 1..., V be a patton of V. Then thee exsts an equpatton Res 1,..., Res T of V such that the followng hold. () Fo all t [T ], [] and v V (G), we have d G,Rest V (v) = 1 T d G,V (v) ± n 2/3, () fo all t [T ], [], we have Res t V = 1 T V ± n 2/3. Lemma Suppose n, s N wth 0 < 1/n ε 1/s 1 and m [n] fo each [2]. Let G be an n-vetex gaph. Suppose that U s a collecton of m 1 subsets of V (G) and U s a collecton of m 2 pas of dsont subsets of V (G) such that each (U 1, U 2 ) U satsfes U 1, U 2 > n 3/4. Let 0 p 1,..., p s 1 wth s =1 p = 1. Then thee exsts a decomposton G 1,..., G s of G satsfyng the followng. () Fo all [s], U U and v V (G), we have d G,U(v) = p d G,U (v) ± n 2/3, () fo all [s] and (U 1, U 2 ) U such that G[U 1, U 2 ] s (ε, d (U1,U 2 ))-egula fo some d (U1,U 2 ), we have that G [U 1, U 2 ] s (2ε, p d (U1,U 2 ))-egula. The followng lemma allows us to fnd well-dstbuted subsets of a collecton of lage sets. The equed sets can be found va a staghtfowad geedy appoach (whle avodng the vetces whch would volate (B3) 3.18 n each step). So we omt the detals. Lemma Suppose n, s, N and 0 < 1/n, 1/s ε d < 1. Let A be a set of sze n, and fo each (, ) [s] [] let A, A be of sze at least dn, and let m, N {0} be such that fo all [s] we have =1 m, εn. Then thee exst sets B 1,1,..., B s, satsfyng the followng. (B1) 3.18 Fo all [s] and [], we have B, A, wth B, = m,, (B2) 3.18 fo all [s] and [], we have B, B, =, (B3) 3.18 fo all v A, we have {(, ) [s] [] : v B, } ε 1/2 s. The followng lemma guaantees a set of -clques n a gaph G whch cove evey vetex a pescbed numbe of tmes. Lemma Let n, m,, t N and 0 < 1/n 1/t σ, 1/ < 1 wth n. Let G be an n-vetex gaph wth δ(g) (1 1 +σ)n. Suppose that fo each v V (G), we have d v [m] {0}. Then thee exsts a mult--gaph H on vetex set V (G) satsfyng the followng. (B1) 3.19 Fo each e E(H), we have G[e] K, (B2) 3.19 fo each v V (G), we have d H (v) d v = (t + 1)m ± 1. Poof. Let m := max {d u d v }. u,v V (G) Then m [m]. Fo a mult-hypegaph H on vetex set V (G) and v V (G), let p H (v) := d H (v) d v. We wll pove that fo each l [m 1] {0}, thee exsts a hypegaph H l satsfyng the followng. (H1) l 3.19 Fo each e E(H), we have G[e] K, (H2) l 3.19 (H l) l(t + 1), (H3) l 3.19 max u,v V (G){p Hl (v) p Hl (u)} m l. Note that H 0 = satsfes (H1) (H3) Assume that fo some l [m 2] {0}, we have aleady constucted H l satsfyng (H1) l 3.19 (H3)l We wll now constuct H l+1. If max u V (G) {p Hl (u)} mn u V (G) {p Hl (u)} 1, then as l m 2, we can let H l+1 := H l, then (H1) l (H3)l hold. Thus assume that max {p H l (u)} mn {p H l (u)} 2. (3.14) u V (G) u V (G)

15 Let A := {v V (G) : p Hl (v) > A BANDWIDTH THEOREM FOR APPROXIMATE DECOMPOSITIONS 15 mn {p H l (u)}} and A max := {v V (G) : p Hl (v) = max {p H l (u)}}. u V (G) u V (G) Fst assume that A. Let A A be a set of at most 1 vetces such that dvdes A + A and p Hl (v) max u A\A p Hl (u) fo all v A. Note that we have ethe A A max o A max A. Then we can tae a collecton A := {A 1,..., A t+1 } of (possbly empty) subsets of A such that the followng hold fo each [t + 1]. A s dvsble by, A A /t +, evey vetex n A belongs to exactly two sets n A and evey vetex n A \ A belongs to exactly one set n A. Now, fo each [t + 1], we have δ(g A ) δ(g) A (1 1/ + σ)n n/t (1 1/ + σ 2/t)n (1 1/)n. Snce V (G)\A contans at most n vetces, and V (G)\A s dvsble by, the Hanal-Szemeéd theoem mples that thee exsts a collecton K of copes of K n G coveng all the vetces n V (G)\A exactly once. Fo each [t + 1], let E := {V (K) : K K }. Then t+1 =1 E coves evey vetex n V (G)\A exactly t + 1 tmes, whle t coves vetces n A \ A exactly t tmes and vetces n A exactly t 1 tmes. Let H l+1 be the mult--gaph on vetex set V (G) wth t+1 E(H l+1 ) := H l E. Then the above constucton wth (H1) l 3.19 mples (H1)l Also (H2)l 3.19 mples that (H l+1) = (H l )+(t+1) (t+1)(l+1), thus (H2) l holds. If A A max, then evey vetex n A max \A s coveed exactly t tmes by t+1 =1 E. Thus, by (3.14), we have max {p H l+1 (u)} = max {p H l (u)} + t and u V (G) u V (G) =1 mn {p H l+1 (u)} = mn {p H l (u)} + t + 1. u V (G) u V (G) If A max A, then evey vetex n A max s coveed exactly t 1 tmes whle evey vetex n A s coveed ethe t 1 tmes o t tmes by t+1 =1 E. Thus, by (3.14), we have max {p H l+1 (u)} = max {p H l (u)} + t 1 and u V (G) u V (G) In both cases, we have mn {p H l+1 (u)} u V (G) mn {p H l (u)} + t. u V (G) { max phl+1 (u) p Hl+1 (v) } max {p H l (u) p Hl (v)} 1 (H3)l 3.19 m l 1. u,v V (G) u,v V (G) Thus (H3) l holds. Next assume that A <. Then we tae two sets B and C n V (G) such that B C = A and B = C =. Then smlaly as befoe, we can tae two collectons E 1 and E 2 of sets of sze such that E 1 coves evey vetex n V (G) \ B exactly once, and E 2 coves evey vetex n V (G) \ C exactly once whle G[e] K fo all e E 1 E 2. Let H l+1 be the mult--gaph wth E(H l+1 ) := H l E 1 E 2. Then, t s easy to see that both (H1) l and (H2)l hold. Also E 1 E 2 coves all vetces n V (G) \ A exactly once o twce, whle t does not cove the vetces n A. Then as befoe, by usng the fact that max u V (G) {p Hl (u)} mn u V (G) {p Hl (u)} 2, we can show that (H3) l holds. Hence, ths shows that thee exsts a hypegaph H m 1 whch satsfes (H1) m (H3)m Let m := max v V (G) {p Hm 1 (v)}. Then (H2) m mples that m (t + 1)m. Also, by (H3) m evey vetex v V (G) satsfes p Hm 1 (v) {m 1, m }. Recall that δ(g) (1 1/)n and dvdes n. Thus the Hanal-Szemeéd theoem guaantees a collecton E of sets of sze whch coves evey vetex of G exactly once, whle G[e] K fo all e E. Thus, by addng all e E to H m 1 exactly (t + 1)m m tmes, we obtan a mult--gaph satsfyng (B1) 3.19 and (B2) 3.19.