Linear connectivity forces large complete bipartite minors
Böhme, 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.
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