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Statements

Subject Item
n2:RIV%2F00216224%3A14330%2F08%3A00024776%21RIV09-GA0-14330___
rdf:type
skos:Concept n19:Vysledek
dcterms:description
Prezentujeme dva přístupy k dosud otevřené domněnce, že průsečíkově kritické grafy mají omezený maximální stupeň. Na pozitivní straně dokazujeme, že tyto grafy nemohou obsahovat subdivizi velkého K_2,n, ale na negativní straně ukazujeme neplatnost domněnky v projektivní rovině. The structure of all known infinite families of crossing--critical graphs has led to the conjecture that crossing--critical graphs have bounded bandwidth. If true, this would imply that crossing--critical graphs have bounded degree, that is, that they cannot contain subdivisions of $K_{1,n}$ for arbitrarily large $n$. In this paper we prove two results that revolve around this conjecture. On the positive side, we show that crossing--critical graphs cannot contain subdivisions of $K_{2,n}$ for arbitrarily large $n$. On the negative side, we show that there are graphs with arbitrarily large maximum degree that are $2$-crossing--critical in the projective plane. The structure of all known infinite families of crossing--critical graphs has led to the conjecture that crossing--critical graphs have bounded bandwidth. If true, this would imply that crossing--critical graphs have bounded degree, that is, that they cannot contain subdivisions of $K_{1,n}$ for arbitrarily large $n$. In this paper we prove two results that revolve around this conjecture. On the positive side, we show that crossing--critical graphs cannot contain subdivisions of $K_{2,n}$ for arbitrarily large $n$. On the negative side, we show that there are graphs with arbitrarily large maximum degree that are $2$-crossing--critical in the projective plane.
dcterms:title
Stars and Bonds in Crossing-Critical Graphs Stars and Bonds in Crossing-Critical Graphs Hvězdy a řezy v průsečíkově kritických grafech
skos:prefLabel
Hvězdy a řezy v průsečíkově kritických grafech Stars and Bonds in Crossing-Critical Graphs Stars and Bonds in Crossing-Critical Graphs
skos:notation
RIV/00216224:14330/08:00024776!RIV09-GA0-14330___
n3:aktivita
n20:P
n3:aktivity
P(GA201/08/0308)
n3:dodaniDat
n16:2009
n3:domaciTvurceVysledku
n13:7595646
n3:druhVysledku
n11:D
n3:duvernostUdaju
n17:S
n3:entitaPredkladatele
n15:predkladatel
n3:idSjednocenehoVysledku
397228
n3:idVysledku
RIV/00216224:14330/08:00024776
n3:jazykVysledku
n10:eng
n3:klicovaSlova
crossing number; crossing-critical graph
n3:klicoveSlovo
n7:crossing-critical%20graph n7:crossing%20number
n3:kontrolniKodProRIV
[E43AFB9045A9]
n3:mistoKonaniAkce
Paris, France
n3:mistoVydani
Nizozemi
n3:nazevZdroje
The International Conference on Topological and Geometric Graph Theory TGGT2008, Electronic Notes in Discrete Mathematics 31, 2008
n3:obor
n18:BA
n3:pocetDomacichTvurcuVysledku
1
n3:pocetTvurcuVysledku
2
n3:projekt
n5:GA201%2F08%2F0308
n3:rokUplatneniVysledku
n16:2008
n3:tvurceVysledku
Hliněný, Petr Salazar, Gelasio
n3:typAkce
n12:WRD
n3:zahajeniAkce
2008-05-19+02:00
s:issn
1571-0653
s:numberOfPages
5
n8:hasPublisher
Elsevier
n14:organizacniJednotka
14330