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1
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On 1-dependent ramsey numbers for graphs

100%
Cockayne E., Mynhardt C.
Discussiones Mathematicae Graph Theory
|
1999
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tom 19
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nr 1
93-110
EN
A set X of vertices of a graph G is said to be 1-dependent if the subgraph of G induced by X has maximum degree one. The 1-dependent Ramsey number t₁(l,m) is the smallest integer n such that for any 2-edge colouring (R,B) of Kₙ, the spanning subgraph B of Kₙ has a 1-dependent set of size l or the subgraph R has a 1-dependent set of size m. The 2-edge colouring (R,B) is a t₁(l,m) Ramsey colouring of Kₙ if B (R, respectively) does not contain a 1-dependent set of size l (m, respectively); in this case R is also called a (l,m,n) Ramsey graph. We show that t₁(4,5) = 9, t₁(4,6) = 11, t₁(4,7) = 16 and t₁(4,8) = 17. We also determine all (4,4,5), (4,5,8), (4,6,10) and (4,7,15) Ramsey graphs.
2
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A lower bound for the irredundance number of trees

100%
Poschen M., Volkmann L.
Discussiones Mathematicae Graph Theory
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2006
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tom 26
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nr 2
209-215
EN
Let ir(G) and γ(G) be the irredundance number and domination number of a graph G, respectively. The number of vertices and leaves of a graph G are denoted by n(G) and n₁(G). If T is a tree, then Lemańska [4] presented in 2004 the sharp lower bound γ(T) ≥ (n(T) + 2 - n₁(T))/3. In this paper we prove ir(T) ≥ (n(T) + 2 - n₁(T))/3. for an arbitrary tree T. Since γ(T) ≥ ir(T) is always valid, this inequality is an extension and improvement of Lemańska's result.
3
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Associative graph products and their independence, domination and coloring numbers

88%
Nowakowski R., Rall D.
Discussiones Mathematicae Graph Theory
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1996
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tom 16
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nr 1
53-79
EN
Associative products are defined using a scheme of Imrich & Izbicki [18]. These include the Cartesian, categorical, strong and lexicographic products, as well as others. We examine which product ⊗ and parameter p pairs are multiplicative, that is, p(G⊗H) ≥ p(G)p(H) for all graphs G and H or p(G⊗H) ≤ p(G)p(H) for all graphs G and H. The parameters are related to independence, domination and irredundance. This includes Vizing's conjecture directly, and indirectly the Shannon capacity of a graph and Hedetniemi's coloring conjecture.
4
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An inequality chain of domination parameters for trees

88%
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.
5
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Domination, independence and irredundance in graphs

77%
Topp J.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS 1. Introduction........................................................................................................ 5  1.1. Purpose and scope................................................................................. 5  1.2. Basic graphtheoretical terms................................................................ 6 2. Domination, independence and irredundance in graphs................................ 9  2.1. Introduction and preliminaries.............................................................. 9  2.2. Domination parameters of vertex and edgedeleted subgraphs..... 15  2.3. Packing and covering numbers............................................................ 25  2.4. Conditions for equalities of domination parameters........................ 35 3. Well covered graphs........................................................................................ 46  3.1. Introduction and preliminary results..................................................... 46  3.2. The well coveredness of products of graphs..................................... 55  3.3. Well covered simplicial and chordal graphs...................................... 67  3.4. Well covered line and total graphs....................................................... 73  3.5. Well covered generalized Petersen graphs........................................ 78  3.6. Well irredundant graphs......................................................................... 80 4. Graphical sequences and sets of integers......................................................... 85  4.1. Dominationfeasible sequences........................................................... 86  4.2. Interpolation properties of domination parameters.......................... 91 References.................................................................................................................... 94
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