Packing Graphs: The packing problem solved

Transcription

1 Packing Graps: Te packing problem solve Yair Caro an Rapael Yuster Department of Matematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification: 05B05,05B40 (primary), 05B30,51E05,94C30,6K05,6K10 (seconary). Submitte: November 8, 1996; Accepte: December, 1996 Deicate to te memory of Paul Erős Abstract For every fixe grap H, we etermine te H-packing number of K n, for all n>n 0 (H). We prove tat if is te number of eges of H, an gc(h) =is te greatest common ivisor of te egrees of H, ten tere exists n 0 = n 0 (H), suc tat for all n>n 0, P(H, K n )= n n 1, unless n =1moan n(n 1)/ = b mo (/) were 1 b, in wic case P (H, K n )= n n 1 1. Our main tool in proving tis result is te eep ecomposition result of Gustavsson. 1 Introuction All graps consiere ere are finite, unirecte an simple. For te stanar grap-teoretic terminology te reaer is referre to [Bo]. Let H be a grap witout isolate vertices. An H- packing of a grap G is a set L = {G 1,...,G s }of ege-isjoint subgraps of G, were eac subgrap zeac603@uvm.aifa.ac.il rapy@mat.tau.ac.il 1

2 te electronic journal of combinatorics 4 (1997), #R1 is isomorpic to H. Te H-packing number of G, enote by P (H, G), is te maximum carinality of an H-packing of G. An H-covering of a grap G is a set L = {G 1,...,G s } of subgraps of G, were eac subgrap is isomorpic to H, were every ege of G appears in at least one member of L. Te H-covering number of G, enote by C(H, G), is te minimum carinality of an H-covering of G. G as an H-ecomposition if it as an H-packing wic is also an H-covering. Te H-packing an H-covering problems are, in general, NP-Complete as sown by Dor an Tarsi [DoTa]. In case G = K n, te H-covering an H-packing problems ave attracte muc attention in te last forty years, an numerous papers were written on tese subjects (cf. [Br95,Ha,MiMu,CoDi,StKaMu] for various surveys). Special cases of tese problems gaine particular interest. 1. P (K k,k n ) wic as been linke to te various Jonson-Sconeim bouns in Coing Teory [BiEt,BrSSlSm,Sc,Jo]. It is known tat P (K k,k n ) is te maximum size of te binary coes of wor-lengt n, constant weigt k, an istance k ork 3. Despite of muc effort only te cases k =3[an k = 4 [] Sc an k =4[]] are solve. Te case k =5is still open [MuYi].. P (C k,k n ) wic is te cycle-system packing problem, solve completely only for k =3,k=4 [an k = 5 [17] ScBi an k =5[17]]. 3. Te packing-covering conjecture for trees saying tat P (T,K n )= ) / an C(T,Kn )= ) / ( is te number of eges of T ) provie n is sufficiently large. Tis conjecture as been prove for all trees on at most 7 vertices [Ro83,Ro93]. A central result concerning H-ecompositions of K n is te teorem of Wilson [Wi] stating tat for sufficiently large n, K n as an H-ecomposition if an only if e(h) ) an gc(h) n 1 were gc(h) is te greatest common ivisor of te egrees of H. Clearly, if te conitions in Wilson s Teorem ol, ten te packing an covering numbers are known. In tis paper we solve all of te conjectures above, for large n, as special consequences of a muc more general result. In fact, for every H, we etermine P (H, K n ), for all n n 0 (H). Teorem 1.1 Let H be a grap wit eges, an let gc(h)=. Ten tere exists n 0 = n 0 (H), suc tat for all n>n 0, P(H, K n )= n n 1, unless n =1moan n(n 1)/ = b mo (/) were 1 b, in wic case P (H, K n )= n n 1 1.

3 te electronic journal of combinatorics 4 (1997), #R1 3 Proof of te main result As mentione in te abstract, our main tool is te following result of Gustavsson [Gu]: Lemma.1 (Gustavsson s Teorem [Gu) ]LetHbe a grap wit eges. Tere exists N = N(H), an ɛ = ɛ(h) > 0, suc tat for all n>n,ifgis a grap on n vertices an m eges, wit δ(g) n(1 ɛ), gc(h) gc(g), an m, ten G as an H-ecomposition. It is wort mentioning tat N(H) in Gustavsson s Teorem is a rater uge constant; in fact, it is a igly exponential function of. A sequence of n positive integers 1... n is calle grapic if tere exists an n-vertex grap wose egree sequence is { 1,..., n }. We sall nee te following teorem of Erős an Gallai [ErGa], wic gives a necessary an sufficient conition for a sequence to be grapic. Lemma. (Erős an Gallai [ErGa) ] Te sequence 1... n of positive integers is grapic if an only if its sum is even an for every t =1,...,n t i t(t 1) + min{t, i }. (1) Proof of Teorem 1.1: Given H, we coose n 0 = n 0 (H) = max{n(h), ɛ(h), 8}, were N(H) an ɛ(h) are as in Lemma.1. Now let n>n 0. Let n 1=amo, were 0 a 1. Let n(n 1 a)/ = b mo (/), were 0 b / 1. Note tat since = gc(h) an is te sum of te egrees of H, / must be an integer. Also note tat (n 1 a)/ is an integer, an so b is well-efine. We sall use te obvious fact tat ( +1)/, since δ(h). Tis means tat n>n 0 8>4 >(a+). Anoter useful fact is tat b + na is even since if is even ten a an n ave ifferent parity, an if is o ten / is even an so if b is o ten a an n are bot o, an if b is even ten eiter n is even or a is even. In te first part of te proof we sall give a lower boun for P (H, K n ), an in te secon part we sall give an upper boun for P (H, K n ), an notice tat te lower an upper bouns coincie. Proving a lower boun for P (H, K n ): We sall first assume tat a 0. Our first goal is to sow te existence of an n-vertex grap wic as b vertices wit egree + a, an n b vertices wit egree a. For tis purpose we sall use Lemma., wit i = a + for i =1,...,b an i = a

4 te electronic journal of combinatorics 4 (1997), #R1 4 for i = b +1,...,n. Notice first tat te sum of te sequence is b + na an tis number is even as mentione above. Let 1 t a +. In tis case, (1) ols since t i t(a + ) =t(t 1) + t(a + t +1) t(t 1)+(a+)(a + 1) = t(t 1)+(a+) (a+)<t(t 1) + n (a + ) t(t 1)+(n t) t(t 1) + min{t, i }. For a + t n we sall prove tat (1) ols by inuction on t, were te base case t = a + was prove above. If t>a+ we use te inuction ypotesis to erive tat t t 1 i = t + i t +(t 1)(t ) + min{t, i } = i=t t + min{t, t } (t 1) + t(t 1) + (a + )+(a+) (a + )+t(t 1) + min{t, i } min{t, i } = t(t 1) + min{t, i }. Tus, tere exists a grap G aving b vertices wit egree + a an n b vertices wit egree a. Consier G = K n \ G. Clearly, gc(g), an G as m eges were m = ( ) n b + na = 1 a) (n(n b))=0mo. Also note tat δ(g) n 1 a = n(1 1+a+ n ) n(1 ɛ(h)), since n>n 0 ɛ(h). Tus, G satisfies te conitions of Lemma.1, an terefore G as an H-ecomposition. Tis means tat P (H, K n ) P(H, G) = m = (n(n 1 a) b)) = n n 1 We now eal wit te case a =0. Ifb= 0 ten K n as an H-ecomposition accoring to Wilson s Teorem [Wi], (or accoring to Lemma.1), so, trivially, P (H, K n )= ) =n n 1 = n n 1 If b>we may elete from K n a subgrap G on b vertices wic is regular (tis is oable since b + na = b is even). As in te case were a 0, te remaining grap G = K n \ G satisfies te conitions of Lemma.1 an terefore ) b n ) P (H, K n ) P(H, G) = = ( = n n 1 = n n 1

5 te electronic journal of combinatorics 4 (1997), #R1 5 Finally, if 1 b ten we can elete from K n a subgrap G on b + vertices wic is regular. Note tat tis can be one since (+1)/ wic implies < +b. Also, if is o ten b an are bot even, so b + is even. Once again, te remaining grap G = K n \ G satisfies te conitions of Lemma.1 an we get P (H, K n ) P(H, G) = ) (b+(/)) ) b n ) = 1= ( 1= n n 1 1. Proving an upper boun for P (H, K n ): Let L be an arbitrary H-packing of K n. Let s enote te carinality of L. Let G enote te ege-union of all te members of L. G contains s eges. Tus G = K n \ G contains ) s eges. Te egree of every vertex in G is 0 mo an so te egree of every vertex in G is a mo. Terefore, te number of eges in G satisfies ( ) n an + c s = for some non-negative integer c. In particular, ) = an+c mo. Tis, in turn, implies tat n(n 1 a)/ = c mo (/). Tus, we must ave c b. Terefore, ) ( an+c n ) an+b s = = n n 1 Since L was an arbitrary H-packing, we ave P (H, K n ) n n 1 Te only remaining case is a = 0 an 1 b. In tis case, we cannot ave c = b. Tis is because every non-isolate vertex of G as egree at least, an terefore tere are at least ( +1)/ eges in G, i.e c/ ( +1)/, wic implies c + 1, but b. We must, terefore ave c b +/. Terefore, s = ) an+c ) an+(b+/) = n n Concluing remarks 1. Teorem 1.1, applie to H = K k yiels, for n n 0 (k), tat P (K k,k n )= n k n 1 k 1, unless k 1 n 1 an n(n 1)/(k 1) mo k is less tan k an greater tan 0, in wic case te above formula soul be reuce by 1. Tis solves, in particular, te relate problem in Coing Teory mentione in te introuction.

6 te electronic journal of combinatorics 4 (1997), #R1 6. Teorem 1.1, applie to H = C k yiels, for n n 0 (k), tat P (C k,k n )= n k n 1 unless n is o an ) =1,mok. 3. If n n 0 (H) an gc(h) = 1, ten P (H, K n )= (n ) e(h). Note tat by first eleting from K n any set of b<e(h) eges were b = ) mo e(h), te remaining grap satisfies te conitions in Gustavsson s Teorem, an since te set of elete eges may be cosen as a subgrap of H we ave C(H, K n )= (n ) e(h), solving, in particular, te packing-covering conjecture for trees. Our approac allows us to solve te covering problem as well. Tis is one in a fortcoming paper [CaYu]. 4 Acknowlegment Te autors wis to tank N. Alon, T. Etzion, R. Mullin, J. Sconeim an Y. Roitty for useful iscussions, elpful information, an sening important references. References [1] S. Bitan an T. Etzion, Te last packing number of quaruples an cyclic SQS, Design, Coes an Cryptograpy 3 (1993), [] A.E. Brouwer, Optimal packing of K 4 s into a K n, J. Combin. Teory, Ser. A 6 (1979), [3] A.E. Brouwer, Block Designs, in: Capter 14 in Hanbook of Combinatorics, R. Graam, M. Grötscel an L. Lovász Es. Elsevier, [4] A. Brouwer, J. Searer, N. Sloane an W. Smit, A new table of constant weigt coes, IEEE Trans. Inform. Teory 36 (1990), [5] B. Bollobás, Extremal Grap Teory, Acaemic Press, [6] Y. Caro an R. Yuster, Covering graps: Te covering problem solve, submitte. [7] C.J. Colbourn an J.H. Dinitz, CRC Hanbook of Combinatorial Design, CRC press 1996.

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