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Linear connectivity forces large complete bipartite minorsBöhme, Thomas ...Dokazano je, da za poljubna naravna števila ▫$a$▫, ▫$s$▫ in ▫$k$▫ obstaja tak ▫$N = N(s,k,a)$▫, da vsak ▫$\frac{31}{2} (a+1)▫$-povezan graf z vsaj ▫$N$▫ vozlišči bodisi vsebuje subdivizijo grafa ... ▫$K_{a,sk}$▫ ali pa minor, ki je izomorfen kopijam grafa ▫$K_{a,k}$▫. Pravzaprav za to zadostuje že povezanost ▫$3a+2$▫ in minimalna stopnja ▫$\frac{31}{2} (a+1) - 3$▫. V posebnem primeru, ko je▫$s=1$▫ in ▫$k=a$▫, sledi, da ima vsak dovolj velik ▫$\frac{31}{2} (a+1)$▫-povezan graf minor ▫$K_a$▫. To je prvi znani rezultat, v katerem linearna povezanost zagotavlja minor ▫$K_a$▫. Ta rezultat tudi posplošuje rezultate Böhmeja in Kostochke in dokazuje hipotezo Fon-Der-Flaassa. Kot dodatno netrivialno posledico dobimo še nov rezultat, ki je v tesni zvezi s Hadwigerjevo domnevo, in rezultat o disjunktnih minorjih ▫$K_{a,k}$▫, ki je podoben izreku Erdős-Pósa za disjunktne cikle v grafu.Source: Journal of combinatorial theory. Series B. - ISSN 0095-8956 (Vol. 99, no. 3, 2009, str. 557-582)Type of material - article, component partPublish date - 2009Language - englishCOBISS.SI-ID - 15145049
Author
Böhme, Thomas |
Kawarabayashi, Ken-ichi |
Maharry, John |
Mohar, Bojan, 1956-
Topics
matematika |
teorija grafov |
minor grafa |
povezanost |
drevesna širina |
Hadwigerjeva hipoteza |
▫$k$▫-prepleteni graf |
grafi na ploskvah |
Eulerjeva formula |
lastnost Erdősa in Póse |
mathematics |
graph theory |
graph minor |
connectivity |
tree-width |
tree-decomposition |
path-decomposition |
complete graph minor |
unavoidable minor |
▫$k$▫-linked |
Hadwiger conjecture |
grid minor |
Vortex structure |
near embedding |
graphs on surfaces |
Euler's formula |
Erdős-Pósa property
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Böhme, Thomas | ![]() |
Kawarabayashi, Ken-ichi | ![]() |
Maharry, John | ![]() |
Mohar, Bojan, 1956- | 01931 |
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