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13.
A theorem about a contractible and light edge
Dvořák, Zdeněk ; Škrekovski, Riste
Leta 1955 je Kotzig pokazal, da vsak ravninski 3-povezan graf vsebuje povezavo stopnje ▫$\le 13$▫. Če pa graf ne vsebuje točke stopnje 3, imamo povezavo stopnje ▫$\le 11$▫. Taki povezavi rečemo lahka ...povezava. Pomemben rezultat Steinitza pravi, da so skeleti konveksnih 3-politopov natanko 3-povezani ravninski grafi. Po drugi strani pa je leta 1961 Tuttle dokazal, da vsak 3-povezan graf vsebuje skrčljivo povezavo. V članku izboljšamo Kotzigov izrek, tako da pokažemo, da vsak tak graf, različen od ▫$K_4$▫, vsebuje povezavo, ki je hkrati lahka in skrčljiva. Kot posledico dobimo, da se da vsak konveksni 3-politop konstruirati iz tetraedra s sekvečno ekspanzijo točk stopnje največ 11.
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