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1
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A note on packing of two copies of a hypergraph

100%
Pilśniak M., Woźniak M.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 1
45-49
EN
A 2-packing of a hypergraph 𝓗 is a permutation σ on V(𝓗) such that if an edge e belongs to 𝓔(𝓗), then σ (e) does not belong to 𝓔(𝓗). We prove that a hypergraph which does not contain neither empty edge ∅ nor complete edge V(𝓗) and has at most 1/2n edges is 2-packable. A 1-uniform hypergraph of order n with more than 1/2n edges shows that this result cannot be improved by increasing the size of 𝓗.
2
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Generalizations of the tree packing conjecture

100%
Gerbner D., Keszegh B., Palmer C.
Discussiones Mathematicae Graph Theory
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2012
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tom 32
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nr 3
569-582
EN
The Gyárfás tree packing conjecture asserts that any set of trees with 2,3,...,k vertices has an (edge-disjoint) packing into the complete graph on k vertices. Gyárfás and Lehel proved that the conjecture holds in some special cases. We address the problem of packing trees into k-chromatic graphs. In particular, we prove that if all but three of the trees are stars then they have a packing into any k-chromatic graph. We also consider several other generalizations of the conjecture.
3
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Packing of nonuniform hypergraphs - product and sum of sizes conditions

100%
Naroski P.
Discussiones Mathematicae Graph Theory
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2009
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tom 29
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nr 3
651-656
EN
Hypergraphs $H₁,...,H_N$ 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.
4
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Calculs de dimensions de packing

80%
Nasr F. B.
Colloquium Mathematicum
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1996
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tom 71
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nr 1
137-148
5
Content available remote

Packing the Hypercube

80%
Offner D.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
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nr 1
85-93
EN
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.
6
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The s-packing chromatic number of a graph

80%
Goddard W., Xu H.
Discussiones Mathematicae Graph Theory
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2012
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tom 32
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nr 4
795-806
EN
Let S = (a₁, a₂, ...) be an infinite nondecreasing sequence of positive integers. An S-packing k-coloring of a graph G is a mapping from V(G) to {1,2,...,k} such that vertices with color i have pairwise distance greater than $a_i$, and the S-packing chromatic number $χ_S(G)$ of G is the smallest integer k such that G has an S-packing k-coloring. This concept generalizes the concept of proper coloring (when S = (1,1,1,...)) and broadcast coloring (when S = (1,2,3,4,...)). In this paper, we consider bounds on the parameter and its relationship with other parameters. We characterize the graphs with $χ_S = 2$ and determine $χ_S$ for several common families of graphs. We examine $χ_S$ for the infinite path and give some exact values and asymptotic bounds. Finally we consider complexity questions, especially about recognizing graphs with $χ_S = 3$.
7
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An inequality chain of domination parameters for trees

70%
Cockayne E., Favaron O., Puech J., Mynhardt C.
Discussiones Mathematicae Graph Theory
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1998
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tom 18
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nr 1
127-142
EN
We prove that the smallest cardinality of a maximal packing in any tree is at most the cardinality of an R-annihilated set. As a corollary to this result we point out that a set of parameters of trees involving packing, perfect neighbourhood, R-annihilated, irredundant and dominating sets is totally ordered. The class of trees for which all these parameters are equal is described and we give an example of a tree in which most of them are distinct.
8
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On the packing of two copies of a caterpillar in its third power

70%
Germain C., Kheddouci H.
Discussiones Mathematicae Graph Theory
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2003
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tom 23
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nr 1
105-115
EN
H. Kheddouci, J.F. Saclé and M. Woźniak conjectured in 2000 that if a tree T is not a star, then there is an edge-disjoint placement of T into its third power.In this paper, we prove the conjecture for caterpillars.
9
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Graphs with Large Generalized (Edge-)Connectivity

61%
Li X., Mao Y.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 4
931-958
EN
The generalized k-connectivity κk(G) of a graph G, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized k-edge-connectivity λk(G). In this paper, graphs of order n such that [...] for even k are characterized.
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