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Packing four copies of a tree into a complete bipartite graph

Liqun Pu, Yuan Tang, Xiaoli Gao (2022)

Czechoslovak Mathematical Journal

In considering packing three copies of a tree into a complete bipartite graph, H. Wang (2009) gives a conjecture: For each tree T of order n and each integer k 2 , there is a k -packing of T in a complete bipartite graph B n + k - 1 whose order is n + k - 1 . We prove the conjecture is true for k = 4 .

Packing of nonuniform hypergraphs - product and sum of sizes conditions

Paweł Naroski (2009)

Discussiones Mathematicae Graph Theory

Hypergraphs H , . . . , H N of order n are mutually packable if one can find their edge disjoint copies in the complete hypergraph of order n. We prove that two hypergraphs are mutually packable if the product of their sizes satisfies some upper bound. Moreover we show that an arbitrary set of the hypergraphs is mutually packable if the sum of their sizes is sufficiently small.

Packing of three copies of a digraph into the transitive tournament

Monika Pilśniak (2004)

Discussiones Mathematicae Graph Theory

In this paper, we show that if the number of arcs in an oriented graph G (of order n) without directed cycles is sufficiently small (not greater than [2/3] n-1), then there exist arc disjoint embeddings of three copies of G into the transitive tournament TTₙ. It is the best possible bound.

Packing Parameters in Graphs

I. Sahul Hamid, S. Saravanakumar (2015)

Discussiones Mathematicae Graph Theory

In a graph G = (V,E), a non-empty set S ⊆ V is said to be an open packing set if no two vertices of S have a common neighbour in G. An open packing set which is not a proper subset of any open packing set is called a maximal open packing set. The minimum and maximum cardinalities of a maximal open packing set are respectively called the lower open packing number and the open packing number and are denoted by ρoL and ρo. In this paper, we present some bounds on these parameters.

Packing the Hypercube

David Offner (2014)

Discussiones Mathematicae Graph Theory

Let G be a graph that is a subgraph of some n-dimensional hypercube Qn. For sufficiently large n, Stout [20] proved that it is possible to pack vertex- disjoint copies of G in Qn so that any proportion r < 1 of the vertices of Qn are covered by the packing. We prove an analogous theorem for edge-disjoint packings: For sufficiently large n, it is possible to pack edge-disjoint copies of G in Qn so that any proportion r < 1 of the edges of Qn are covered by the packing.

Packing Trees Into n-Chromatic Graphs

András Gyárfás (2014)

Discussiones Mathematicae Graph Theory

We show that if a sequence of trees T1, T2, ..., Tn−1 can be packed into Kn then they can be also packed into any n-chromatic graph.

Paired- and induced paired-domination in {E,net}-free graphs

Oliver Schaudt (2012)

Discussiones Mathematicae Graph Theory

A dominating set of a graph is a vertex subset that any vertex belongs to or is adjacent to. Among the many well-studied variants of domination are the so-called paired-dominating sets. A paired-dominating set is a dominating set whose induced subgraph has a perfect matching. In this paper, we continue their study. We focus on graphs that do not contain the net-graph (obtained by attaching a pendant vertex to each vertex of the triangle) or the E-graph (obtained by attaching...

Partitions of networks that are robust to vertex permutation dynamics

Gary Froyland, Eric Kwok (2015)

Special Matrices

Minimum disconnecting cuts of connected graphs provide fundamental information about the connectivity structure of the graph. Spectral methods are well-known as stable and efficient means of finding good solutions to the balanced minimum cut problem. In this paper we generalise the standard balanced bisection problem for static graphs to a new “dynamic balanced bisection problem”, in which the bisecting cut should be minimal when the vertex-labelled graph is subjected to a general sequence of vertex...

Partitions of some planar graphs into two linear forests

Piotr Borowiecki, Mariusz Hałuszczak (1997)

Discussiones Mathematicae Graph Theory

A linear forest is a forest in which every component is a path. It is known that the set of vertices V(G) of any outerplanar graph G can be partitioned into two disjoint subsets V₁,V₂ such that induced subgraphs ⟨V₁⟩ and ⟨V₂⟩ are linear forests (we say G has an (LF, LF)-partition). In this paper, we present an extension of the above result to the class of planar graphs with a given number of internal vertices (i.e., vertices that do not belong to the external face at a certain fixed embedding of...

Path and cycle factors of cubic bipartite graphs

M. Kano, Changwoo Lee, Kazuhiro Suzuki (2008)

Discussiones Mathematicae Graph Theory

For a set S of connected graphs, a spanning subgraph F of a graph is called an S-factor if every component of F is isomorphic to a member of S. It was recently shown that every 2-connected cubic graph has a {Cₙ | n ≥ 4}-factor and a {Pₙ | n ≥ 6}-factor, where Cₙ and Pₙ denote the cycle and the path of order n, respectively (Kawarabayashi et al., J. Graph Theory, Vol. 39 (2002) 188-193). In this paper, we show that every connected cubic bipartite graph has a {Cₙ | n ≥ 6}-factor, and has a {Pₙ | n...

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