Packing and Covering Dense Graphs Noga Alon ∗ Yair Caro y Raphael Yuster z Abstract Let d be a positive integer. A graph G is called d-divisible if d divides the degree of each vertex of G. G is called nowhere d-divisible if no degree of a vertex of G is divisible by d. For a graph H, gcd(H) denotes the greatest common divisor of the degrees of the vertices of H. The H-packing number of G is the maximum number of pairwise edge disjoint copies of H in G. The H-covering number of G is the minimum number of copies of H in G whose union covers all edges of G. Our main result is the following: For every fixed graph H with gcd(H) = d, there exist positive constants (H) and N(H) such that if G is a graph with at least N(H) vertices and has minimum degree at least (1 − (H)) G , then the H-packing number of G and the H-covering number of G can be computed j j in polynomial time. Furthermore, if G is either d-divisible or nowhere d-divisible, then there is a closed formula for the H-packing number of G, and the H-covering number of G. Further extensions and solutions to related problems are also given. 1 Introduction All graphs considered here are finite, undirected and simple, unless otherwise noted. For the standard graph-theoretic terminology the reader is referred to [1]. Let H be a graph without isolated vertices. An H-covering of a graph G is a set L = G1;:::;Gs of subgraphs of G, where f g each subgraph is isomorphic to H, and 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. An H-packing of a graph G is a set L = G1;:::;Gs of edge-disjoint subgraphs of G, where each f g 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. G has an H-decomposition if it has an H-packing which is ∗Department of Mathematics, Tel-Aviv University, Israel, e-mail: [email protected]. Research supported in part by a USA Israeli BSF grant and by a grant from the Israel Science Foundation. yDepartment of Mathematics, University of Haifa-ORANIM, Tivon 36006, Israel. e-mail: [email protected] zDepartment of Mathematics, University of Haifa-ORANIM, Tivon 36006, Israel. e-mail: [email protected] 1 also an H-covering. Recently, exact formulas for P (H; Kn) and C(H; Kn) have been obtained, for n n(H) [3, 4]. A main tool which is used in both these papers is the result of Gustavsson [10] ≥ concerning the decomposition of dense graphs. In case the graph G is not complete, it is known that computing P (H; G) and C(H; G) is, in general, NP-Hard, as shown by Dor and Tarsi [7]. The purpose of this paper is to extend Gustavsson's result, and the above mentioned results for Kn and show that if the graph G is very dense, then the H-packing and H-covering numbers of G can be determined in polynomial time, and in many cases, these numbers can be given by a closed formula. To describe our results we need the following definitions. A graph G is called d-divisible if the degree of each vertex of G is a multiple of d. G is called nowhere d-divisible if no vertex of G has degree which is a multiple of d. Note that regular graphs are either d-divisible or nowhere d-divisible for every d. Also note that every graph is 1-divisible. For a graph H let gcd(H) denote the greatest common divisor of the degrees of the vertices of H. For example, gcd(C4) = 2, whereas gcd(T ) = 1 for every tree T . Our main results are the following: Theorem 1.1 (Packing Dense Graphs) Let H be a graph with h edges, and let gcd(H) = d. Then there exist N = N(H), and = (H) such that if G = (V; E) is a graph with n > N(H) vertices and δ(G) > (1 (H))n, then P (H; G) can be determined in polynomial time. Furthermore, − if G is either d-divisible or nowhere d-divisible then, v V αv P (H; G) = P 2 ; b 2h c where αv is the degree of vertex v, rounded down to the closest multiple of d. The r.h.s. of this formula should be reduced by 1 if G is d-divisible and 0 < E mod h d2=2. j j ≤ Theorem 1.2 (Covering Dense Graphs) Let H be a graph with h edges, and let gcd(H) = d. Then there exist N = N(H), and = (H) such that if G = (V; E) is a graph with n > N(H) vertices and δ(G) > (1 (H))n, then C(H; G) can be determined in O(n2:5) time. Furthermore, − if G is either d-divisible or nowhere d-divisible then, v V αv C(H; G) = P 2 ; d 2h e where αv is the degree of vertex v, rounded up to the closest multiple of d. The r.h.s. of this formula should be increased by 1 if G is d-divisible and h divides E + d=2. j j This paper is organized as follows. In section 2 we describe the tools and prove some lemmas which will be used in the proofs of Theorems 1.1 and 1.2. In section 3 we prove Theorem 1.1. In section 4 we prove Theorem 1.2. The final section contains results and extensions which solve some 2 variations of the packing and covering problems, among them are the leave and excess problems (see, e.g. [6] pages 263-264 and pages 411-412, and [11, 12]), and the efficient 2-overlap covering problem of Etzion [5, 2]. Throughout this sequel we use the notation e(G) to denote the number of edges of a graph G. dG(v) denotes the degree of vertex v in G. For X V , G[X] denotes the subgraph of G = (V; E) ⊂ induced by X. 2 Preparing the tools As mentioned in the introduction, 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 N0 = N0(H), and γ = γ(H) > 0, such that for all n > N0, if G is a graph on n vertices and m edges, with δ(G) n(1 γ), gcd(H) gcd(G), and h m, then G has an H-decomposition. 2 ≥ − j j It is worth mentioning that N0(H) in Gustavsson's Theorem is a rather huge constant; in fact, it 24 1 is a highly exponential function of h. Also, the γ(H) is very small; in fact it is less than 10− h− . Thus, the graph G needs to be large and dense, but it may still be far from being complete (i.e., Ω(n2) edges may be missing). The following lemma is a cornerstone in the proofs of Theorems 1.1 and 1.2. In essence, it shows that dense graphs contain sparse spanning subgraphs with predetermined degrees. In fact, this lemma can be viewed as an approximate bounded-degree version of the f-Factor Theorem of Tutte [13]. Lemma 2.2 Let d be a positive integer, and let G = (V; E) be a graph on n vertices with δ(G) ≥ n n=(4d + 2) + 2d + 1. Let νv v V be a set of positive integers not exceeding 2d whose sum − f j 2 g is even. Then G contains a spanning subgraph G∗ in which the degree of each vertex v is νv. i i Proof: For i = 1;:::; 2d + 1, let A be a (0-1)-sequence indexed by V , where A = 1 if i νv, v ≤ i 1 2d+1 and Av = 0 if i > νv. Clearly, A has all its elements equal to 1, while A has all its elements equal to 0. We call a sequence Ai odd if it contains an odd number of ones. Consider all the odd sequences Ai for 2 i 2d, having at most n=2 elements equal to 1. We may pick from each such ≤ ≤ sequence one location which is zero, such that no two sequences picked the same location. This can be done since the number of sequences is at most 2d, while the number of zeroes in each is at least n=2, and n=2 > 2d. We modify the location picked for Ai to 1, and the corresponding location of A1 is set to 0. Now consider all the odd sequences Ai for 2 i 2d, having more ≤ ≤ than n=2 elements equal to 1. We may pick from each such sequence one location which equals 1, 3 such that no two sequences picked the same location. We modify the location picked for Ai to 0, and the corresponding location of A2d+1 is set to 1. This process guarantees that all the sequences A2;:::;A2d have an even number of ones. If A1 also has an even number of ones after the process is completed, then so does A2d+1.