Wyniki wyszukiwania - Biblioteka Nauki

1

Coloring Some Finite Sets in ℝn

100%

Balogh J. , Kostochka A. , Raigorodskii A.

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nr 1

25-31

EN

This note relates to bounds on the chromatic number χ(ℝn) of the Euclidean space, which is the minimum number of colors needed to color all the points in ℝn so that any two points at the distance 1 receive different colors. In [6] a sequence of graphs Gn in ℝn was introduced showing that . For many years, this bound has been remaining the best known bound for the chromatic numbers of some lowdimensional spaces. Here we prove that and find an exact formula for the chromatic number in the case of n = 2k and n = 2k − 1.

2

On the Independence Number of Edge Chromatic Critical Graphs

100%

Pang S. , Miao L. , Song W. , Miao Z.

Discussiones Mathematicae Graph Theory

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2014

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tom 34

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nr 3

577-584

EN

In 1968, Vizing conjectured that for any edge chromatic critical graph G = (V,E) with maximum degree △ and independence number α (G), α (G) ≤ [...] . It is known that α (G) < [...] |V |. In this paper we improve this bound when △≥ 4. Our precise result depends on the number n2 of 2-vertices in G, but in particular we prove that α (G) ≤ [...] |V | when △ ≥ 5 and n2 ≤ 2(△− 1)

3

On the independence number of some strong products of cycle-powers

100%

Jurkiewicz M. , Kubale M. , Ocetkiewicz K.

Foundations of Computing and Decision Sciences

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2015

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tom Vol. 40, No. 2

133--141

EN

In the paper we give some theoretical and computational results on the third strong power of cycle-powers, for example, we have found the independence numbers α((C210)⊠3) = 30 and α((C414)⊠3) = 14. A number of optimizations have been introduced to improve the running time of our exhaustive algorithm used to establish the independence number of the third strong power of cycle-powers. Moreover, our results establish new exact values and/or lower bounds on the Shannon capacity of noisy channels.

4

A note of arbitrarily vertex decomposable graphs

100%

Marczyk A.

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tom Vol. 26, no. 1

109-118

EN

A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n1,..., nk) of positive integers such that n1 + ... + nk = n there exists a partition (V1,..., Vk) of the vertex set of G such that for each i ∈ {1,..., k}, Vi induces a connected subgraph of G on ni vertices. In this paper we show that if G is a two-connected graph on n vertices with the independence number at most ⌈n/2⌉ and such that the degree sum of any pair of non-adjacent vertices is at least n - 3, then G is arbitrarily vertex decomposable. We present another result for connected graphs satisfying a similar condition, where the bound n - 3 is replaced by n - 2.

5

Looseness and Independence Number of Triangulations on Closed Surfaces

88%

Nakamoto A. , Negami S. , Ohba K. , Suzuki Y.

Discussiones Mathematicae Graph Theory

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2016

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tom 36

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nr 3

545-554

EN

The looseness of a triangulation G on a closed surface F2, denoted by ξ (G), is defined as the minimum number k such that for any surjection c : V (G) → {1, 2, . . . , k + 3}, there is a face uvw of G with c(u), c(v) and c(w) all distinct. We shall bound ξ (G) for triangulations G on closed surfaces by the independence number of G denoted by α(G). In particular, for a triangulation G on the sphere, we have [...] and this bound is sharp. For a triangulation G on a non-spherical surface F2, we have ξ (G) ≤ 2α(G) + l(F2) − 2, where l(F2) = [(2 − χ(F2))/2] with Euler characteristic χ(F2).

6

On Generalized Sierpiński Graphs

88%

Rodríguez-Velázquez J. A. , Rodríguez-Bazan E. D. , Estrada-Moreno A.

Discussiones Mathematicae Graph Theory

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2017

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tom 37

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nr 3

547-560

EN

In this paper we obtain closed formulae for several parameters of generalized Sierpiński graphs S(G, t) in terms of parameters of the base graph G. In particular, we focus on the chromatic, vertex cover, clique and domination numbers.

7

Bounds on the 2-domination number in cactus graphs

88%

Chellali M.

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tom Vol. 26, no. 1

5-12

EN

A 2-dominating set of a graph G is a set D of vertices of G such that every vertex not in S is dominated at least twice. The minimum cardinality of a 2-dominating set of G is the 2-domination number γ2(G). We show that if G is a nontrivial connected cactus graph with k(G) even cycles (k(G) ≥ 0), then γ2(G) ≥ γt(G) - k(G), and if G is a graph of order n with at most one cycle, then γ2(G) ≥ (n + l - s)/2 improving Fink and Jacobson's lower bound for trees with l > s, where γt(G), l and s are the total domination number, the number of leaves and support vertices of G, respectively. We also show that if T is a tree of order n ≥ 3, then γ2(T) ≤ β(T) + s - 1, where β(T) is the independence number of T.

8

On the Total Graph of Mycielski Graphs, Central Graphs and Their Covering Numbers

63%

Patil H. P. , Pandiya Raj R.

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nr 2

361-371

EN

The technique of counting cliques in networks is a natural problem. In this paper, we develop certain results on counting of triangles for the total graph of the Mycielski graph or central graph of star as well as completegraph families. Moreover, we discuss the upper bounds for the number of triangles in the Mycielski and other well known transformations of graphs. Finally, it is shown that the achromatic number and edge-covering number of the transformations mentioned above are equated.

9

Extending the MAX Algorithm for Maximum Independent Set

63%

Lê N. C. , Brause C. , Schiermeyer I.

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tom 35

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nr 2

365-386

EN

The maximum independent set problem is an NP-hard problem. In this paper, we consider Algorithm MAX, which is a polynomial time algorithm for finding a maximal independent set in a graph G. We present a set of forbidden induced subgraphs such that Algorithm MAX always results in finding a maximum independent set of G. We also describe two modifications of Algorithm MAX and sets of forbidden induced subgraphs for the new algorithms.

10

The Quest for A Characterization of Hom-Properties of Finite Character

51%

Broere I. , Matsoha M. D. V. , Heidema J.

Discussiones Mathematicae Graph Theory

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2016

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tom 36

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nr 2

479-500

EN

A graph property is a set of (countable) graphs. A homomorphism from a graph G to a graph H is an edge-preserving map from the vertex set of G into the vertex set of H; if such a map exists, we write G → H. Given any graph H, the hom-property →H is the set of H-colourable graphs, i.e., the set of all graphs G satisfying G → H. A graph property P is of finite character if, whenever we have that F ∈ P for every finite induced subgraph F of a graph G, then we have that G ∈ P too. We explore some of the relationships of the property attribute of being of finite character to other property attributes such as being finitely-induced-hereditary, being finitely determined, and being axiomatizable. We study the hom-properties of finite character, and prove some necessary and some sufficient conditions on H for →H to be of finite character. A notable (but known) sufficient condition is that H is a finite graph, and our new model-theoretic proof of this compactness result extends from hom-properties to all axiomatizable properties. In our quest to find an intrinsic characterization of those H for which →H is of finite character, we find an example of an infinite connected graph with no finite core and chromatic number 3 but with hom-property not of finite character.