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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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The representation of multi-hypergraphs by set intersections

100%
Bylka S., Komar J.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
|
nr 3
565-582
EN
This paper deals with weighted set systems (V,𝓔,q), where V is a set of indices, $𝓔 ⊂ 2^V$ and the weight q is a nonnegative integer function on 𝓔. The basic idea of the paper is to apply weighted set systems to formulate restrictions on intersections. It is of interest to know whether a weighted set system can be represented by set intersections. An intersection representation of (V,𝓔,q) is defined to be an indexed family $R = (R_v)_{v∈ V}$ of subsets of a set S such that $|⋂_{v∈ E} R_v| = q(E)$ for each E ∈ 𝓔. A necessary condition for the existence of such representation is the monotonicity of q on 𝓔 i.e., if F ⊂ 𝓔 then q(F) ≥ q(𝓔). Some sufficient conditions for weighted set systems representable by set intersections are given. Appropriate existence theorems are proved by construction of the solutions. The notion of intersection multigraphs to intersection multi- hypergraphs - hypergraphs with multiple edges, is generalized. Some conditions for intersection multi-hypergraphs are formulated.
2
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On the intersection graphs of ideals of direct product of rings

70%
Rad N., Jafari S., Ghosh S.
Discussiones Mathematicae - General Algebra and Applications
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2014
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tom 34
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nr 2
191-201
EN
In this paper we first calculate the number of vertices and edges of the intersection graph of ideals of direct product of rings and fields. Then we study Eulerianity and Hamiltonicity in the intersection graph of ideals of direct product of commutative rings.
3
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Tree-Like Partial Hamming Graphs

70%
Gologranc T.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
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nr 1
137-150
EN
Tree-like partial cubes were introduced in [B. Brešar, W. Imrich, S. Klavžar, Tree-like isometric subgraphs of hypercubes, Discuss. Math. Graph Theory, 23 (2003), 227-240] as a generalization of median graphs. We present some incorrectnesses from that article. In particular we point to a gap in the proof of the theorem about the dismantlability of the cube graph of a tree-like partial cube and give a new proof of that result, which holds also for a bigger class of graphs, so called tree-like partial Hamming graphs. We investigate these graphs and show some results which imply previously-known results on tree-like partial cubes. For instance, we characterize tree-like partial Hamming graphs and prove that every tree-like partial Hamming graph G contains a Hamming graph that is invariant under every automorphism of G. The latter result is a direct consequence of the result about the dismantlability of the intersection graph of maximal Hamming graphs of a tree-like partial Hamming graph.
4
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A Finite Characterization and Recognition of Intersection Graphs of Hypergraphs with Rank at Most 3 and Multiplicity at Most 2 in the Class of Threshold Graphs

70%
Metelsky Y., Schemeleva K., Werner F.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 1
13-28
EN
We characterize the class [...] L32 $L_3^2 $ of intersection graphs of hypergraphs with rank at most 3 and multiplicity at most 2 by means of a finite list of forbidden induced subgraphs in the class of threshold graphs. We also give an O(n)-time algorithm for the recognition of graphs from [...] L32 $L_3^2 $ in the class of threshold graphs, where n is the number of vertices of a tested graph.
5
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On θ-graphs of partial cubes

70%
Klavžar S., Kovse M.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 2
313-321
EN
The Θ-graph Θ(G) of a partial cube G is the intersection graph of the equivalence classes of the Djoković-Winkler relation. Θ-graphs that are 2-connected, trees, or complete graphs are characterized. In particular, Θ(G) is complete if and only if G can be obtained from K₁ by a sequence of (newly introduced) dense expansions. Θ-graphs are also compared with familiar concepts of crossing graphs and τ-graphs.
6
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Intersection graph of gamma sets in the total graph

61%
Chelvam T., Asir T.
Discussiones Mathematicae Graph Theory
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2012
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tom 32
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nr 2
341-356
EN
In this paper, we consider the intersection graph $I_{Γ}(ℤₙ)$ of gamma sets in the total graph on ℤₙ. We characterize the values of n for which $I_{Γ}(ℤₙ)$ is complete, bipartite, cycle, chordal and planar. Further, we prove that $I_{Γ}(ℤₙ)$ is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivity and some domination parameters of $I_{Γ}(ℤₙ)$.
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