Packing of graphs with small product of sizes

Transcription

1 Joural of Combatoral Theory, Seres B 98 (008) Note Packg of graphs wth small product of szes Alexadr V. Kostochka a,b,,gexyu c, a Departmet of Mathematcs, Uversty of Illos, Urbaa, IL 680, USA b Sobolev Isttute of Mathematcs, Novosbrsk , Russa c Departmet of Mathematcs, Vaderblt Uversty, Nashvlle, TN 3740, USA Receved 6 August 007 Avalable ole 8 March 008 Abstract We show that for every ɛ>0, there exsts 0 = 0 (ɛ) such that for every > 0,two-vertex graphs G ad G wth e(g )e(g ) ( ɛ) pack, uless they belog to a well-defed famly of exceptos. Ths exteds a well-kow result by Sauer ad Specer. 008 Elsever Ic. All rghts reserved. Keywords: Graph packg. Itroducto How may edges should a -vertex graph have to cota every graph wth at most vertces ad at most m edges? Erdős ad Stoe [7] proved that for every postve teger d ad postve c ad suffcetly large, every graph G of order wth at least ( /)( /d) + c edges cotas a complete (d + )-partte graph wth t vertces each part, where t teds to fty wth. It follows that ths G cotas every d-colorable graph o t vertces, ad, partcular, that G cotas every graph wth less tha ( d+) edges. Later Bollobás, Erdős, ad Smoovts [] showed that t a log /(d log(/c)) for some postve costat a ad cojectured that ths ca be mproved as follows: t b log / log(/c). Chvátal ad Szemeréd [6] verfed ths cojecture by provg the followg theorem. E-mal addresses: kostochk@math.uuc.edu (A.V. Kostochka), gex.yu@vaderblt.edu (G. Yu). Research supported part by NSF grat DMS ad RFBR grat Research supported part by NSF grat DMS /$ see frot matter 008 Elsever Ic. All rghts reserved. do:0.06/j.jctb

2 4 A.V. Kostochka, G. Yu / Joural of Combatoral Theory, Seres B 98 (008) 4 45 Theorem. (See Chvátal ad Szemeréd [6].) For each postve teger d ad each c>0, there s a 0 = 0 (d, c) such that for each 0, every graph of order wth at least ( /)( /d) + c edges cotas a complete (d + )-partte graph wth at least (log )/(500 log(/c)) vertces each part. Bollobás ad Eldrdge [] cosdered also Turá-type codtos guarateeg that a -vertex graph cotas every subgraph wth α edges for α</. They proved a boud ths drecto the laguage of packg ad posed a cojecture whch was proved by Bradt [5]. Recall that two graphs pack f oe of the graphs s cotaed the complemet of the other. The ext theorem s a somewhat smplfed verso of Bradt s result. Theorem. (See Bradt [5].) For every 0 <α</, there exsts 0 = 0 (α) such that f > 0, e(g ) α, ad e(g ) 3 α 3/, the G ad G pack. Bollobás, Kostochka, ad Nakprast [3] exteded Theorem to the case α. A smplfed verso of t s as follows. Theorem 3. (See Bollobás, Kostochka ad Nakprast [3].) Let / α<. Let G ad G be graphs of order >( 40 α )6 such that e(g ) α, e(g ) 3 3/, ad Δ(G )<. The G α( α) ad G pack. Sauer ad Specer [8] proved the followg boud terms of the product of the szes of graphs. Theorem 4. (See Sauer ad Specer [8].) Two -vertex graphs G ad G pack, f ( ) e(g )e(g )<. The followg examples of graphs that do ot pack show that the codto e(g )e(g )< ( ) caot be weakeed wthout troducg other restrctos. Example. G = K ad G = K K. Example. G = K, ad G has o solated vertces. Note that Example, f s eve ad G s a perfect matchg, the e(g )e(g ) = ( ). Also ote that e(g ) + e(g ) ca be aroud 3/. Bollobás ad Eldrdge [] proved that ths may happe oly f oe of the graphs has a all-adjacet vertex or s small. I a bt smplfed form, ther result s as follows. Theorem 5. (See Bollobás ad Eldrdge [].) Let G ad G be graphs of order >0 such that Δ(G ), Δ(G ) ad e(g ) + e(g ) 3. The G ad G pack. Ths boud s also sharp as the followg examples show. Example 3. G = G = K 3 K, 4.

3 A.V. Kostochka, G. Yu / Joural of Combatoral Theory, Seres B 98 (008) Example 4. G = K, K ad G s -regular. Example 5. G = K, 3 K, s dvsble by 3, ad G = K 3 K 3. Teo ad Yap [9] showed that for 3, Examples 3 5 are the oly pars (G,G ) of -vertex graphs wth e(g ) + e(g ) = that do ot pack. I ths paper we stregthe Theorem 4 by descrbg (for large ) the pars (G,G ) of -vertex graphs wth e(g )e(g ) ( ɛ) that do ot pack. Theorem 6. For every ɛ>0, there exsts N, such that for all >N,ftwo-vertex graphs G ad G wth e(g )e(g ) ( ɛ) () do ot pack, the oe of the followg holds: () oe of the graphs s K ad the other has exactly oe edge; or () oe of the graphs has maxmum degree ad the other has mmum degree at least oe; or () oe of the graphs s a tragle, ad the other has depedece umber two. Observe that there are expoetally may pars (G,G ) of -vertex graphs satsfyg () or () wth e(g )e(g ) 0.9. Although -vertex graphs wth depedece umber two ad fewer tha ( ɛ) /3 edges may have a complcated structure, we ca polyomal tme check ay graph whether t possesses ths property. We beleve that t wll be suffcetly harder to descrbe the pars (G,G ) of -vertex graphs wth e(g )e(g ) ( + ɛ) that do ot pack eve for small postve ɛ. Note that Examples 3 5 fall to ths category. Yet aother example s as follows. Example 6. G = K 4, s dvsble by 3, ad G = K /3 K /3 K /3. I the proof of Theorem 6 we wll make use of the followg fact. Theorem 7. (See Bollobás, Kostochka ad Nakprast [4].) Let d. Let G be a d-degeerate graph of order ad maxmum degree Δ ad G a graph of order ad maxmum degree at most Δ.If40Δ l Δ <ad 40dΔ <, the G ad G pack. Recall that a graph s d-degeerate f every subgraph of t cotas a vertex of degree at most d.. Proof of Theorem 6 Fx a 0 <ɛ<0.. Let be large. Suppose that Theorem 6 does ot hold for ɛ ad,.e. that there are -vertex graphs G ad G satsfyg () that do ot pack ad do ot belog to the famles descrbed by () (). We may assume that e(g ) e(g ). So, by (), e(g )< ɛ < ( ɛ/). Letα = e(g )/. By above, 0 <α< ɛ/. ()

4 44 A.V. Kostochka, G. Yu / Joural of Combatoral Theory, Seres B 98 (008) 4 45 Let Δ, =,, deote the maxmum degree of G. By Theorems ad 3, f e(g )< 3 3 ad Δ < α, the G ad G pack. So we cosder the followg two cases: () Δ α ad e(g )< 3 3, ad () e(g ) 3 3. Case. Δ α ad e(g )< 3 3.Letw V(G ) be a vertex of maxmum degree Δ G. If e(g )<( ɛ/)/, the sce s large ad e(g )< 3 3, G ad G pack by Theorem. So we assume that e(g ) ( ɛ/)/. Note that the by (), e(g ) ( ɛ). If G has a solated vertex, say w, the let G = G w, ad G = G w. Wehave e(g ) = e(g ) ad e ( G ) = e(g ) Δ(G ) ( ɛ) + <( ɛ/) (3) α for >( ɛ( α) ). By () ad (3), for such ad {, },wehave Δ ( G ( ) ) e G <( ɛ/) ( ). By Theorem 5, G ad G pack. Thus G ad G pack as well (by placg w at w). Assume ow that G has o solated vertces. Sce () does ot hold, G has o vertex of degree. Sce every coected graph H cotag a cycle has E(H) V(H) ad e(g )<( ɛ/), G has at least ɛ tree compoets. So there s a tree compoet T of G wth at most ɛ/ = ɛ vertces. We wll frst place o the vertces of G the vertces of T, ad the fd a placemet of the remag vertces. Let t = V(T) ad let the vertces of T be ordered u,u,...,u t such a way that u s a leaf ad for every =,...,t,vertexu has exactly oe eghbor {u,u,...,u }. Place u at w. Sce d G (w) <, we may place u at a o-eghbor w of w G.Let G = G u ad G = G w. Suppose ow that t ad we have already placed u,...,u o vertces w,...,w of G. By the orderg of V(T), u + has exactly oe eghbor u j {u,u,...,u }. Observe that e ( G ) e(g ) Δ <( ɛ/). (4) Therefore, w has at least ( ɛ/) o-eghbors G.Atmost t ɛ of these vertces are already occuped by u,...,u. Thus for large, there s a o-eghbor w + of w ot yet occuped. Place there u +. Ths way, we place all vertces V(T) o vertces of G wthout coflcts. Let G = G V(T)ad G = G {w,w,...,w t }. If we fd a packg of G wth G, the we obta a packg of G wth G, a cotradcto. By (4), for large, e ( G ) ( ( ) + e G ) ( ɛ/) (t ) + ( ɛ/) ( t) 3. By () ad (4), for =,, Δ(G ) e(g )<( ɛ/) (G ), ad hece ether G or G has a all-adjacet vertex. Thus by Theorem 5, G ad G pack. Case. e(g ) 3 3. The ( ) e(g ) ( ɛ) / 3 3 = 3( ɛ). (5) Sce e(g )<3, G has at least 6 >/ solated vertces. Let v,v,...,v be a orderg of V(G ) such that v has maxmum degree G [v,...,v ].LetG =

5 A.V. Kostochka, G. Yu / Joural of Combatoral Theory, Seres B 98 (008) G [v / +,...,v ] ad Δ be the maxmum degree of G. The 0 e(g ) Δ e(g ) Δ ( + )/, ad thus Δ e(g )/. LetG be the graph obtaed after removg / solated vertces from G.Let = /. If G s a forest, the let d =. Otherwse, let d be the maxmum postve teger such that G s d-degeerate. Sce e(g ) d(d + )/, d e(g ). (6) By (5), 40Δ l Δ < 40e(G ) l <. Sce G ad G do ot pack, Theorem 7 yelds 40dΔ.So / 40dΔ 40 e(g ) e(g ) 80 e(g ) ( ɛ) e(g ) = 80 ( ɛ). e(g ) That s, e(g ) (60 ( ɛ)) < 0 5.Letc 0 = e(g ).Ifc 0 = ad G ad G do ot pack, the G = K ad hece () holds. If c 0 = ad G ad G do ot pack, the the complemet G of G s cotaed ether a matchg (f G s a -path), or K, or K 3 (f G has two solated edges). I all cases, G has at least ( ) edges. Therefore, e(g )e(g ) 3, a cotradcto to (). The case G = K 3 ad G ad G do ot pack s the other way to express (). So, we have 3 c ad G K 3. Hece the sze of the complemet G of G s at least ( ) ) ( ɛ) = ( c0 + ɛc0 c 0 ) ( c0 + ɛ. c 0 By Theorem, G cotas complete ( 0.5c log 0 +)-partte graph wth t 500 log(c 0 /ɛ) > 05 vertces each part. Thus, f χ ( G ) + 0.5c0, (7) the G cotas G,.e., G ad G pack ad hece G ad G pack. Ths s certaly the case f c 0 = 3 ad G K 3.Ifc 0 {4, 5}, the χ(g ) 3 ad + 0.5c 0 =3. Smlarly, f c 0 {6, 7}, the χ(g ) 4 ad + 0.5c 0 =4. Let c 0 8 ad k = χ(g ). Sce G s d-degeerate, (6) yelds k + d + c 0.But for each real c 0 8, we have c 0 0.5c 0 ad so (7) holds. Ths proves the theorem. Refereces [] B. Bollobás, S.E. Eldrdge, Packgs of graphs ad applcatos to computatoal complexty, J. Comb. Theory Ser. B 5 (978) [] B. Bollobás, P. Erdős, M. Smoovts, O the structure of edge graphs. II, J. Lodo Math. Soc. () (975/976) 9 4. [3] B. Bollobás, A.V. Kostochka, K. Nakprast, O two cojectures o packg of graphs, Comb. Probab. Comput. 4 (005) [4] B. Bollobás, A.V. Kostochka, K. Nakprast, Packg d-degeerate graphs, J. Comb. Theory Ser. B 98 (008) [5] S. Bradt, A extremal result for subgraphs wth few edges, J. Comb. Theory Ser. B 64 (995) [6] V. Chvátal, E. Szemeréd, O the Erdős Stoe theorem, J. Lodo Math. Soc. () 3 (98) [7] P. Erdős, A.H. Stoe, O the structure of lear graphs, Bull. Amer. Math. Soc. 5 (946) [8] N. Sauer, J. Specer, Edge dsjot placemet of graphs, J. Comb. Theory Ser. B 5 (978) [9] S.K. Teo, H.P. Yap, Packg two graphs of order havg total sze at most, Graphs Comb. 6 (990)