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On cyclic edge-connectivity of fullerenes
Klavdija Kutnar, Dragan Marušič, 2008, original scientific article

Abstract: A graph is said to be cyclically ▫$k$▫-edge-connected, if at least ▫$k$▫ edges must be removed to disconnect it into two components, each containing a cycle. Such a set of ▫$k$▫ edges is called a cyclic-k-edge cutset and it is called a trivial cyclic-k-edge cutset if at least one of the resulting two components induces a single ▫$k$▫-cycle. It is known that fullerenes, that is, 3-connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclically 5-edge-connected. In this article it is shown that a fullerene ▫$F$▫ containing a nontrivial cyclic-5-edge cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces whose neighboring faces are also pentagonal. Moreover, it is shown that ▫$F$▫ has a Hamilton cycle, and as a consequence at least ▫$15 \cdot 2^{n/20-1/2}$▫ perfect matchings, where ▫$n$▫ is the order of ▫$F$▫.
Keywords: graph, fullerene graph, cyclic edge-connectivity, hamilton cycle, perfect matching
Published in RUP: 03.04.2017; Views: 3937; Downloads: 146
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The price of connectivity for cycle transversals
Tatiana Romina Hartinger, Matthew Johnson, Martin Milanič, Daniël Paulusma, 2015, published scientific conference contribution

Keywords: price of connectivity, hereditary graph class, path, cycle, transversal
Published in RUP: 08.08.2016; Views: 5013; Downloads: 129
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Vector connectivity in graphs
Martin Milanič, Endre Boros, Ferdinando Cicalese, Pinar Heggernes, Pim Van 't Hof, Romeo Rizzi, 2014, published scientific conference contribution abstract (invited lecture)

Keywords: vektorska povezanost, aproksimacijski algoritmi, polinomski algoritmi, NP-poln problem, razcepljeni grafi, kografi, drevesa, vector connectivity, approximation algorithms, poynomial time algorithms, NP-hard problem, split graphs, cographs, trees
Published in RUP: 15.10.2015; Views: 5718; Downloads: 348
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On the connectivity of bipartite distance-balanced graphs
Štefko Miklavič, Primož Šparl, 2012, original scientific article

Abstract: A connected graph ▫$\varGamma$▫ is said to be distance-balanced whenever for any pair of adjacent vertices ▫$u,v$▫ of ▫$\varGamma$▫ the number of vertices closer to ▫$u$▫ than to ▫$v$▫ is equal to the number of vertices closer to ▫$v$▫ than to ▫$u$▫. In [K. Handa, Bipartite graphs with balanced ▫$(a,b)$▫-partitions, Ars Combin. 51 (1999), 113-119] Handa asked whether every bipartite distance-balanced graph, that is not a cycle, is 3-connected. In this paper the Handa question is answered in the negative. Moreover, we show that a minimal bipartite distance-balanced graph, that is not a cycle and is not 3-connected, has 18 vertices and is unique. In addition, we give a complete classification of non-3-connected bipartite distance-balanced graphs for which the minimal distance between two vertices in a 2-cut is three. All such graphs are regular and for each ▫$k \geq 3$▫ there exists an infinite family of such graphs which are ▫$k$▫-regular.Furthermore, we determine a number of structural properties that a bipartite distance-balanced graph, which is not 3-connected, must have. As an application, we give a positive answer to the Handa question for the subfamily of bipartite strongly distance-balanced graphs.
Keywords: graph theory, connected graphs, connectivity, distance-balanced graphs, bipartite graphs
Published in RUP: 15.10.2013; Views: 10742; Downloads: 105
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