Packings by cliques and by finite families of graphs (original) (raw)
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Packing Graphs: The packing problem solved
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 ,
Ars Combinatoria, 1998
We deal with the concept of packings in graphs, which may be regarded as ageneralization of the theory of graph design. In particular we construct a vertexandedge-disjoint packing of Kn (wheren2 mod 4 equals 0 or 1) with edges of differentcyclic length. Moreover we consider edge-disjoint packings in complete graphs withuniform linear forests (and the resulting packings have special additional
Operations Research Letters, 1982
For a fixed family ¡ of graphs, an ¡ -packing of a graph ¢ is a set of pairwise vertex-disjoint (or edge-disjoint) subgraphs of ¢ , each isomorphic to an element of ¡ . We focus on the algorithmic aspects of the problem of finding an ¡ -packing that maximizes the number of covered edges. We present results for
Induced Graph Packing Problems
Graphs and Combinatorics, 2010
Let H be a set of undirected graphs. The induced H-packing problem in an input graph G is to find a subgraph Q of G of maximum size such that each connected component of Q is an induced subgraph of G and is isomorphic to some member of H. In this paper we focus on the case when H consists of factor-critical graphs and a certain family of 'propellers'. Clarifying the methods developed in the related theory of non-induced graph packings, we show a Gallai-Edmonds type structure theorem and a Berge-Tutte type minimax formula. We also give an Edmonds type alternating forest algorithm for the case when H consists of a sequential set of stars and factor-critical graphs. This simplifies the related result of Egawa, Kano and Kelmans.
A list version of graph packing
Discrete Mathematics, 2016
We consider the following generalization of graph packing. Let G 1 = (V 1 , E 1) and G 2 = (V 2 , E 2) be graphs of order n and G 3 = (V 1 ∪V 2 , E 3) a bipartite graph. A bijection f from V 1 onto V 2 is a list packing of the triple (G 1 , G 2 , G 3) if uv ∈ E 2 implies f (u)f (v) / ∈ E 2 and vf (v) / ∈ E 3 for all v ∈ V 1. We extend the classical results of Sauer and Spencer and Bollobás and Eldridge on packing of graphs with small sizes or maximum degrees to the setting of list packing. In particular, we extend the well-known Bollobás-Eldridge Theorem, proving that if ∆(G 1) ≤ n−2, ∆(G 2) ≤ n−2, ∆(G 3) ≤ n−1, and |E 1 |+|E 2 |+|E 3 | ≤ 2n−3, then either (G 1 , G 2 , G 3) packs or is one of 7 possible exceptions. Hopefully, the concept of list packing will help to solve some problems on ordinary graph packing, as the concept of list coloring did for ordinary coloring.
Journal of Graph Theory, 2003
Let G be a xed collection of digraphs. Given a digraph H , a Gpacking of H is a collection of vertex disjoint subgraphs of H , each isomorphic to a member of G. A G-packing, P, is maximum if the number of vertices belonging to some member of P is maximum, over all G-packings. The analogous problem for undirected graphs has been extensively studied in the literature. We concentrate on the cases when G is a family of paths. We show G-packing is NP-complete when (essentially) G is not one of the families fP 1 g, or fP 1 ;P 2 g: When G = fP 1 g, the G-packing problem is simply the matching problem. The main focus of our paper is the case when G = fP 1 ;P 2 g, the directed paths of length one and two. We present a collection of augmenting con gurations such that a packing is maximum if and only if it contains no augmenting con guration. We also present a min-max condition which yields a concise certi cate for maximality of packings. We apply these results to obtain a polynomial time algorithm for nding maximum fP 1 ;P 2 g-packings in arbitrary digraphs.
Combinatorica, 1992
G and H, two simple graphs, can be packed if G is isomorphic to a subgraph of H, the complement of H. A theorem of Catlin, Spencer and Sauer gives a sufficient condition for the existence of packing in terms of the product of the maximal degrees of G and H. We improve this theorem for bipartite graphs. Our condition involves products of a maximum degree with an average degree. Our relaxed condition still guarantees a packing of the two bipartite graphs.
On packing and covering numbers of graphs
Discrete Mathematics, 1991
In this paper we present a characterization of connected graphs of order (k + 1)n with k-covering number n, a characterization of trees in which the k-packing and k-covering numbers are the same, and we prove that the smallest number of subgraphs most k which cover the vertices of a block graph equals the k-packing number. of diameter at * This research was partially supported by the Heinrich Hertz Foundation and it was carried out while the author was visiting the Technical University of Aachen.
A nice class for the vertex packing problem
Discrete Applied Mathematics, 1997
A class W* of graphs for which the vertex packing problem can be solved in polynomial time is described. Graphs in V* can be obtained from bipartite graphs and claw-free graphs by repeated substitutions. A forbidden subgraphs characterization of the class V' is given.
Packing problems in edge-colored graphs
Discrete Applied Mathematics, 1994
Let F be a fixed edge-colored graph. We consider the problem of packing the greatest possible number of vertex disjoint copies of F into a given complete edge-colored graph. We observe that this problem is NP-hard unless F consists of isolated vertices and edges or unless there are only two colors and F is properly 2-edge-colored.