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Nordhaus-Gaddum-type theorems for decompositions into many parts
Füredi, Zoltán ...
▫$k$▫-terica ▫$(G_1,...,G_k)$▫ vpetih podgrafov grafa ▫$G$▫ je ▫$k$▫-dekompozicija, če množice povezav teh podgrafov tvorijo razbitje množice ▫$E(G)$▫. Izrek Nordhausa in Gadduna poda meje za ...▫$\chi(G_1) + \chi(G_2)$▫ in ▫$\chi(G_1) \chi(G_2)$▫ za vse 2-dekompozicije grafa ▫$G$▫. Za grafovski parameter ▫$p$▫ naj bo ▫$p(k;G)$▫ maksimum vsote ▫$\sum_{i=1}^{k}p(G_i)$▫ čez vse ▫$k$▫-dekompozicije grafa ▫$G$▫. Za klično število ▫$\omega$▫, kromatično število ▫$\chi$▫, seznamsko število ▫$\chi_{\ell}$▫ ter Szekeres-Wilfovo število ▫$\sigma$▫ velja ▫$\omega(2;K_n) = \chi(2;K_n) = \chi_\ell (2;K_n) = \sigma(2;K_n) = n + 1$▫. V članku so podane zgornje in spodnje meje za ▫$\omega(k;K_n)$▫, ▫$\chi(k;K_n)$▫, ▫$\chi_\ell(k;K_n)$▫ in ▫$\sigma(k;K_n)$▫. Zadnje tri se obnašajo različno za velike ▫$k$▫. Podani sta tudi zgornja in spodnja meja za maksimum ▫$\chi(k;G)$▫ čez vse grafe vložene na dano ploskev.
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