Packing Graphs:

The packing problem solved

Yair Caro ^{∗} and Raphael Yuster ^{†} Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification: 05B05,05B40 (primary), 05B30,51E05,94C30,62K05,62K10 (secondary).

Submitted: November 28, 1996; Accepted: December 2, 1996

Dedicated to the memory of Paul Erdős

Abstract

For every fixed graph H, we determine the H-packing number of K_{n}, for all n > n_{0}(H). We prove that if h is the number of edges of H, and gcd(H) = d is the greatest common divisor of the degrees of H, then there exists n_{0} = n_{0}(H), such that for all n > n_{0},

P (H, K_{n}) = ⌊\frac{dn}{2h}⌊\frac{n − 1}{d}⌋⌋,

unless n = 1 mod d and n(n − 1)/d = b mod (2h/d) where 1 ≤ b ≤ d, in which case

P (H, K_{n}) = ⌊\frac{dn}{2h}⌊\frac{n − 1}{d}⌋⌋ − 1.

Our main tool in proving this result is the deep decomposition result of Gustavsson.

1 Introduction

All graphs considered here are finite, undirected and simple. For the standard graph-theoretic terminology the reader is referred to [5]. Let H be a graph without isolated vertices. An H-packing the electronic journal of combinatorics 4 (1997), #R1 2 of a graph G is a set L = {G_{1}, …, G_{s}} of edge-disjoint subgraphs of G, where each subgraph is isomorphic to H. The H-packing number of G, denoted by P (H, G), is the maximum cardinality of an H-packing of G. An H-covering of a graph G is a set L = {G_{1}, …, G_{s}} of subgraphs of G, where each subgraph is isomorphic to H, where every edge of G appears in at least one member of L. The H-covering number of G, denoted by C(H, G), is the minimum cardinality of an H-covering of G. G has an H-decomposition if it has an H-packing which is also an H-covering. The H-packing and H-covering problems are, in general, NP-Complete as shown by Dor and Tarsi [8]. In case G = K_{n}, the H-covering and H-packing problems have attracted much attention in the last forty years, and numerous papers were written on these subjects (cf. [3, 11, 13, 7, 20] for various surveys). Special cases of these problems gained particular interest.

∗e-mail: zeac603@uvm.haifa.ac.il †e-mail: raphy@math.tau.ac.il

1. P (K_{k}, K_{n}) which has been linked to the various Johnson-Schonheim bounds in Coding Theory [1, 4, 18, 12]. It is known that P (K_{k}, K_{n}) is the maximum size of the binary codes of wordlength n, constant weight k, and distance 2k − 2 or 2k − 3. Despite of much effort only the cases k = 3 [18] and k = 4 [2] are solved. The case k = 5 is still open [14].

2. P (C_{k}, K_{n}) which is the cycle-system packing problem, solved completely only for k = 3, k = 4 [19] and k = 5 [17].

3. The packing-covering conjecture for trees saying that P (T, K_{n}) = ⌊\binom{n}{2}/h⌋ and C(T, K_{n}) = ⌈\binom{n}{2}/h⌉ (h is the number of edges of T) provided n is sufficiently large. This conjecture has been proved for all trees on at most 7 vertices [15, 16].

A central result concerning H-decompositions of K_{n} is the theorem of Wilson [21] stating that for sufficiently large n, K_{n} has an H-decomposition if and only if e(H) | \binom{n}{2} and gcd(H) | n − 1 where gcd(H) is the greatest common divisor of the degrees of H. Clearly, if the conditions in Wilson’s Theorem hold, then the packing and covering numbers are known.

In this paper we solve all of the conjectures above, for large n, as special consequences of a much more general result. In fact, for every H, we determine P (H, K_{n}), for all n ≥ n_{0}(H).

Theorem 1.1 Let H be a graph with h edges, and let gcd(H)=d. Then there exists n_{0} = n_{0}(H), such that for all n > n_{0},

P (H, K_{n}) = ⌊\frac{dn}{2h}⌊\frac{n − 1}{d}⌋⌋,

unless n = 1 mod d and n(n − 1)/d = b mod (2h/d) where 1 ≤ b ≤ d, in which case

P (H, K_{n}) = ⌊\frac{dn}{2h}⌊\frac{n − 1}{d}⌋⌋ − 1.

2 Proof of the main result

As mentioned in the abstract, our main tool is the following result of Gustavsson [10]:

Lemma 2.1 (Gustavsson’s Theorem [10]) Let H be a graph with h edges. There exists N = N(H), and ϵ = ϵ(H) > 0, such that for all n > N, If G is a graph on n vertices and m edges, with δ(G) ≥ n(1 − ϵ), gcd(H) | gcd(G), and h | m, then G has an H-decomposition. □

It is worth mentioning that N(H) in Gustavsson’s Theorem is a rather huge constant; in fact, it is a highly exponential function of h.

A sequence of n positive integers d_{1} ≥ d_{2} ≥ … ≥ d_{n} is called graphic if there exists an n-vertex graph whose degree sequence is {d_{1}, …, d_{n}}. We shall need the following theorem of Erdős and Gallai [9], which gives a necessary and sufficient condition for a sequence to be graphic.

Lemma 2.2 (Erdős and Gallai [9]) The sequence d_{1} ≥ d_{2} ≥ … ≥ d_{n} of positive integers is graphic if and only if its sum is even and for every t = 1, …, n

∑_{i=1}^{t} d_{i} ≤ t(t − 1) + ∑_{i=t+1}^{n} min{t, d_{i}}.  (1)

□

Proof of Theorem 1.1: Given H, we choose n_{0} = n_{0}(H) = max\{N(H), \frac{2h}{ϵ(H)}, 8h\}, where N(H) and ϵ(H) are as in Lemma 2.1. Now let n > n_{0}. Let n − 1 = a mod d, where 0 ≤ a ≤ d − 1. Let n(n − 1 − a)/d = b mod (2h/d), where 0 ≤ b ≤ 2h/d − 1. Note that since d = gcd(H) and 2h is the sum of the degrees of H, 2h/d must be an integer. Also note that (n − 1 − a)/d is an integer, and so b is well-defined. We shall use the obvious fact that h ≥ d(d + 1)/2, since δ(H) ≥ d. This means that

n > n_{0} ≥ 8h > 4d^{2} > (a + d)^{2}.

Another useful fact is that bd + na is even since if d is even then a and n have different parity, and if d is odd then 2h/d is even and so if b is odd then a and n are both odd, and if b is even then either n is even or a is even. In the first part of the proof we shall give a lower bound for P (H, K_{n}), and in the second part we shall give an upper bound for P (H, K_{n}), and notice that the lower and upper bounds coincide. Proving a lower bound for P (H, K_{n}): We shall first assume that a ≠ 0. Our first goal is to show the existence of an n-vertex graph which has b vertices with degree d + a, and n − b vertices with degree a. For this purpose we shall use Lemma 2.2, with d_{i} = a + d for i = 1, …, b and d_{i} = a for i = b + 1, …, n. Notice first that the sum of the sequence is bd + na and this number is even as mentioned above. Let 1 ≤ t ≤ a + d. In this case, (1) holds since

∑_{i=1}^{t} d_{i} ≤ t(a + d) = t(t − 1) + t(a + d − t + 1) ≤ t(t − 1) + (a + d)(a + d − 1) =

t(t − 1) + (a + d)^{2} − (a + d) < t(t − 1) + n − (a + d) ≤ t(t − 1) + (n − t) ≤ t(t − 1) + ∑_{i=t+1}^{n} min{t, d_{i}}.

For a + d ≤ t ≤ n we shall prove that (1) holds by induction on t, where the base case t = a + d was proved above. If t > a + d we use the induction hypothesis to derive that

∑_{i=1}^{t} d_{i} = d_{t} + ∑_{i=1}^{t−1} d_{i} ≤ d_{t} + (t − 1)(t − 2) + ∑_{i=t}^{n} min{t, d_{i}} =

d_{t} + min{t, d_{t}} − 2(t − 1) + t(t − 1) + ∑_{i=t+1}^{n} min{t, d_{i}}

≤ (a + d) + (a + d) − 2(a + d) + t(t − 1) + ∑_{i=t+1}^{n} min{t, d_{i}} = t(t − 1) + ∑_{i=t+1}^{n} min{t, d_{i}}.

Thus, there exists a graph G^{∗} having b vertices with degree d + a and n − b vertices with degree a. Consider G = K_{n} \ G^{∗}. Clearly, d | gcd(G), and G has m edges where

m = \binom{n}{2} − \frac{bd + na}{2} = \frac{d}{2}(\frac{n(n − 1 − a)}{d} − b)) = 0 mod h.

Also note that δ(G) ≥ n − 1 − a − d = n(1 − \frac{1+a+d}{n}) ≥ n(1 − ϵ(H)), since n > n_{0} ≥ \frac{2h}{ϵ(H)}. Thus, G satisfies the conditions of Lemma 2.1, and therefore G has an H-decomposition. This means that

P (H, K_{n}) ≥ P (H, G) = \frac{m}{h} = \frac{d}{2h}(\frac{n(n − 1 − a)}{d} − b)) = ⌊\frac{dn}{2h}⌊\frac{n − 1}{d}⌋⌋.

We now deal with the case a = 0. If b = 0 then K_{n} has an H-decomposition according to Wilson’s Theorem [21], (or according to Lemma 2.1), so, trivially,

P (H, K_{n}) = \frac{\binom{n}{2}}{h} = \frac{dn}{2h} \frac{n − 1}{d} = ⌊\frac{dn}{2h}⌊\frac{n − 1}{d}⌋⌋.

If b > d we may delete from K_{n} a subgraph G^{∗} on b vertices which is d regular (this is doable since bd + na = bd is even). As in the case where a ≠ 0, the remaining graph G = K_{n} \ G^{∗} satisfies the conditions of Lemma 2.1 and therefore

P (H, K_{n}) ≥ P (H, G) = \frac{\binom{n}{2} − \frac{bd}{2}}{h} = ⌊\frac{\binom{n}{2}}{h}⌋ = ⌊\frac{dn}{2h} \frac{n − 1}{d}⌋ = ⌊\frac{dn}{2h}⌊\frac{n − 1}{d}⌋⌋.

the electronic journal of combinatorics 4 (1997), #R1 5 Finally, if 1 ≤ b ≤ d then we can delete from K_{n} a subgraph G^{∗} on b + \frac{2h}{d} vertices which is d regular. Note that this can be done since h ≥ d(d+1)/2 which implies d ≤ \frac{2h}{d} < \frac{2h}{d} +b. Also, if d is odd then b and \frac{2h}{d} are both even, so b + \frac{2h}{d} is even. Once again, the remaining graph G = K_{n} \ G^{∗} satisfies the conditions of Lemma 2.1 and we get

P (H, K_{n}) ≥ P (H, G) = \frac{\binom{n}{2} − \frac{(b+(2h/d))d}{2}}{h} = \frac{\binom{n}{2} − \frac{bd}{2}}{h} − 1 = ⌊\frac{\binom{n}{2}}{h}⌋ − 1 = ⌊\frac{dn}{2h}⌊\frac{n − 1}{d}⌋⌋ − 1.

Proving an upper bound for P (H, K_{n}): Let L be an arbitrary H-packing of K_{n}. Let s denote the cardinality of L. Let G denote the edge-union of all the members of L. G contains sh edges. Thus G^{∗} = K_{n} \ G contains \binom{n}{2} − sh edges. The degree of every vertex in G is 0 mod d and so the degree of every vertex in G^{∗} is a mod d. Therefore, the number of edges in G^{∗} satisfies

\binom{n}{2} − sh = \frac{an + cd}{2}

for some non-negative integer c. In particular, \binom{n}{2} = \frac{an+cd}{2} mod h. This, in turn, implies that n(n − 1 − a)/d = c mod (2h/d). Thus, we must have c ≥ b. Therefore,

s = \frac{\binom{n}{2} − \frac{an+cd}{2}}{h} ≤ \frac{\binom{n}{2} − \frac{an+bd}{2}}{h} = ⌊\frac{dn}{2h}⌊\frac{n − 1}{d}⌋⌋.

Since L was an arbitrary H-packing, we have

P (H, K_{n}) ≤ ⌊\frac{dn}{2h}⌊\frac{n − 1}{d}⌋⌋.

The only remaining case is a = 0 and 1 ≤ b ≤ d. In this case, we cannot have c = b. This is because every non-isolated vertex of G^{∗} has degree at least d, and therefore there are at least d(d + 1)/2 edges in G^{∗}, i.e cd/2 ≥ d(d + 1)/2, which implies c ≥ d + 1, but b ≤ d. We must, therefore have c ≥ b + 2h/d. Therefore,

s = \frac{\binom{n}{2} − \frac{an+cd}{2}}{h} ≤ \frac{\binom{n}{2} − \frac{an+(b+2h/d)d}{2}}{h} = ⌊\frac{dn}{2h}⌊\frac{n − 1}{d}⌋⌋ − 1.

□

3 Concluding remarks

1. Theorem 1.1, applied to H = K_{k} yields, for n ≥ n_{0}(k), that

P (K_{k}, K_{n}) = ⌊\frac{n}{k}⌊\frac{n − 1}{k − 1}⌋⌋,

unless k −1 | n−1 and n(n−1)/(k −1) mod k is less than k and greater than 0, in which case the above formula should be reduced by 1. This solves, in particular, the related problem in Coding Theory mentioned in the introduction.

2. Theorem 1.1, applied to H = C_{k} yields, for n ≥ n_{0}(k), that

P (C_{k}, K_{n}) = ⌊\frac{n}{k}⌊\frac{n − 1}{2}⌋⌋

unless n is odd and \binom{n}{2} = 1, 2 mod k.

3. If n ≥ n_{0}(H) and gcd(H) = 1, then P (H, K_{n}) = ⌊\frac{\binom{n}{2}}{e(H)}⌋. Note that by first deleting from K_{n} any set of b < e(H) edges where b = \binom{n}{2} mod e(H), the remaining graph satisfies the conditions in Gustavsson’s Theorem, and since the set of deleted edges may be chosen as a subgraph of H we have C(H, K_{n}) = ⌈\frac{\binom{n}{2}}{e(H)}⌉, solving, in particular, the packing-covering conjecture for trees.

Our approach allows us to solve the covering problem as well. This is done in a forthcoming paper [6].

4 Acknowledgment

The authors wish to thank N. Alon, T. Etzion, R. Mullin, J. Schonheim and Y. Roditty for useful discussions, helpful information, and sending important references.

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