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1
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Requiring that Minimal Separators Induce Complete Multipartite Subgraphs

100%
McKee T. A.
Discussiones Mathematicae Graph Theory
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2017
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tom 38
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nr 1
263-273
EN
Complete multipartite graphs range from complete graphs (with every partite set a singleton) to edgeless graphs (with a unique partite set). Requiring minimal separators to all induce one or the other of these extremes characterizes, respectively, the classical chordal graphs and the emergent unichord-free graphs. New theorems characterize several subclasses of the graphs whose minimal separators induce complete multipartite subgraphs, in particular the graphs that are 2-clique sums of complete, cycle, wheel, and octahedron graphs.
2
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On An Extremal Problem In The Class Of Bipartite 1-Planar Graphs

100%
Czap J., Przybyło J., Škrabuľáková E.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 1
141-151
EN
A graph G = (V, E) is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite 1-planar graphs with prescribed numbers of vertices in partite sets. Bipartite 1-planar graphs are known to have at most 3n − 8 edges, where n denotes the order of a graph. We show that maximal-size bipartite 1-planar graphs which are almost balanced have not significantly fewer edges than indicated by this upper bound, while the same is not true for unbalanced ones. We prove that the maximal possible size of bipartite 1-planar graphs whose one partite set is much smaller than the other one tends towards 2n rather than 3n. In particular, we prove that if the size of the smaller partite set is sublinear in n, then |E| = (2 + o(1))n, while the same is not true otherwise.
3
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A Note on the Seven Bridges of Königsberg Problem

100%
Naumowicz A.
Formalized Mathematics
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2014
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tom 22
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nr 2
177-178
EN
In this paper we account for the formalization of the seven bridges of Königsberg puzzle. The problem originally posed and solved by Euler in 1735 is historically notable for having laid the foundations of graph theory, cf. [7]. Our formalization utilizes a simple set-theoretical graph representation with four distinct sets for the graph’s vertices and another seven sets that represent the edges (bridges). The work appends the article by Nakamura and Rudnicki [10] by introducing the classic example of a graph that does not contain an Eulerian path. This theorem is item #54 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.
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

84%
Metelsky Y., Schemeleva K., Werner F.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
|
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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Sharp Upper Bounds on the Clar Number of Fullerene Graphs

84%
Gao Y., Zhang H.
Discussiones Mathematicae Graph Theory
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2017
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tom 38
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nr 1
155-163
EN
The Clar number of a fullerene graph with n vertices is bounded above by ⌊n/6⌋ − 2 and this bound has been improved to ⌊n/6⌋ − 3 when n is congruent to 2 modulo 6. We can construct at least one fullerene graph attaining the upper bounds for every even number of vertices n ≥ 20 except n = 22 and n = 30.
6
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The classification of partially symmetric 3-braid links

84%
Stoimenov A.
Open Mathematics
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2015
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tom 13
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nr 1
EN
We classify 3-braid links which are amphicheiral as unoriented links, including a new proof of Birman- Menasco’s result for the (orientedly) amphicheiral 3-braid links. Then we classify the partially invertible 3-braid links.
7
Content available remote

Nanonetworks: The graph theory framework for modeling nanoscale systems

61%
Živkovic J., Tadic B.
Nanoscale Systems: Mathematical Modeling, Theory and Applications
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2013
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tom 2
30-48
EN
Nanonetwork is defined as a mathematical model of nanosize objects with biological, physical and chemical attributes, which are interconnected within certain dynamical process. To demonstrate the potentials of this modeling approach for quantitative study of complexity at nanoscale, in this survey, we consider three kinds of nanonetworks: Genes of a yeast are connected by weighted links corresponding to their coexpression along the cell cycle; Gold nanoparticles, arranged on a substrate, are linked via quantum tunneling junctions which enable single-electron conduction; A network of similar profiles of force–distance curves consists of sequences of states of a molecular complex from HIV–1 virus observed in repeated single-molecule force spectroscopy experiments. The graph-theory analysis of these systems reveals their organizational principles, quantifies the relation between the function of nanostructured materials and their architecture, and helps understand the character of fluctuations at nanoscale.
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