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 e-mail: zeac603@uvm.aifa.ac.il e-mail: rapy@mat.tau.ac.il 1
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.
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
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
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 1 1. 3 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.
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), 83-313. [] A.E. Brouwer, Optimal packing of K 4 s into a K n, J. Combin. Teory, Ser. A 6 (1979), 78-97. [3] A.E. Brouwer, Block Designs, in: Capter 14 in Hanbook of Combinatorics, R. Graam, M. Grötscel an L. Lovász Es. Elsevier, 1995. [4] A. Brouwer, J. Searer, N. Sloane an W. Smit, A new table of constant weigt coes, IEEE Trans. Inform. Teory 36 (1990), 1334-1380. [5] B. Bollobás, Extremal Grap Teory, Acaemic Press, 1978. [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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