Colorings, Crossings, and Cliques
نویسندگان
چکیده
For r = 7, 8 we prove that if χ(G) = r, then cr(G) ≥ cr(Kr). The case r = 5 of the Weak Hajós Conjecture is equivalent to the Four Color Theorem, and the case r = 6 follows from a theorem of Oporowski and Zhao. We present partial results when r = 9.
منابع مشابه
Crossings, Colorings, and Cliques
Albertson conjectured that if graph G has chromatic number r, then the crossing number of G is at least that of the complete graph Kr. This conjecture in the case r = 5 is equivalent to the four color theorem. It was verified for r = 6 by Oporowski and Zhao. In this paper, we prove the conjecture for 7 ≤ r ≤ 12 using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on...
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