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1.
The minor crossing number of graphs with an excluded minor
Drago Bokal, Gašper Fijavž, David Richard Wood, 2008, izvirni znanstveni članek

Opis: The minor crossing number of a graph ▫$G$▫ is the minimum crossing number of a graph that contains ▫$G$▫ as a minor. It is proved that for every graph ▫$H$▫ there is a constant ▫$c$▫, such that every graph ▫$G$▫ with no ▫$H$▫-minor has minor crossing number at most ▫$c|V(G)|$▫.
Ključne besede: mathematics, graph theory, graph minor, excluded minor, crossing number, minor crossing number
Objavljeno: 10.07.2015; Ogledov: 270; Prenosov: 42
.pdf Celotno besedilo (189,15 KB)
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2.
General lower bounds for the minor crossing number of graphs
Drago Bokal, Éva Czabarka, László Székely, Imrich Vrt'o, 2008

Opis: There are three general lower bound techniques for the crossing numbers of graphs: the Crossing Lemma, the bisection method and the embedding method. Inthis contribution, we present their adaptations to the minor crossing number. Using the adapted bounds, we improve on the known bounds on the minor crossing number of hypercubes. We also point out relations of the minor crossing number to string graphs.
Ključne besede: teorija grafov, prekrižno število, minor, hiper kocke, graph theory, minor crossing number, graph minor, string graphs, hypercubes
Objavljeno: 10.07.2015; Ogledov: 266; Prenosov: 39
URL Povezava na celotno besedilo

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General lower bounds for the minor crossing number of graphs
Drago Bokal, Éva Czabarka, László Székely, Imrich Vrt'o, 2010, izvirni znanstveni članek

Opis: Obstajajo tri splošne spodnje meje za minorsko prekrižno število grafov: prekrižna lema, metoda z bisekcijo in metoda z vložitvami. V tem prispevku predstavimo njihove prilagoditve za minorsko prekrižno število grafov. S tako pridobljenimi spodnjimi mejami izboljšamo znane rezultate za minorsko prekrižno število hiperkock. Poleg navedenih rezultatov predstavimo tudi povezavo med minorskim prekrižnim številom in grafi, predstavljivimi s krivuljami (string-graphs).
Ključne besede: teorija grafov, minorsko prekrižno število, grafovski minor, krivuljski grafi, hiper kocke, graph theory, minor crossing number, graph minor, string graphs, hypercubes
Objavljeno: 10.07.2015; Ogledov: 319; Prenosov: 60
URL Povezava na celotno besedilo

5.
Crossing number additivity over edge cuts
Drago Bokal, Markus Chimani, Jesús Leanõs, 2013, izvirni znanstveni članek

Opis: Consider a graph ▫$G$▫ with a minimal edge cut ▫$F$▫ and let ▫$G_1$▫, ▫$G_2$▫ be the two (augmented) components of ▫$G-F$▫. A long-open question asks under which conditions the crossing number of ▫$G$▫ is (greater than or) equal to the sum ofcthe crossing numbers of ▫$G_1$▫ and ▫$G_2$▫ - which would allow us to consider those graphs separately. It is known that crossing number is additive for ▫$|F| in {0,1,2}$▫ and that there exist graphs violating this property with ▫$|F| ge 4$▫. In this paper, we show that crossing number is additive for ▫$|F|=3$▫, thus closing the final gap in the question. The techniques generalize to show that minor crossing number is additive over edge cuts of arbitrary size, as well as to provide bounds for crossing number additivity in arbitrary surfaces. We point out several applications to exact crossing number computation and crossing-critical graphs, as well as provide a very general lower bound for the minor crossing number of the Cartesian product of an arbitrary graph with a tree.
Ključne besede: matematika, teorija grafov, prekrižno število, minor, mathematics, graph theory, crossing number, minor
Objavljeno: 10.07.2015; Ogledov: 423; Prenosov: 43
URL Povezava na celotno besedilo

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