Realizing Lattice Points in 3-Space as the Chromatic Numbers of Three Factors of a Complete Graph

摘要

If {G_1, G_2, G_3} is a factorization of the complete graph of order n, then it is known that χ(G_1) + χ{G_1) + χ(G_3) - 3 ≤ n ≤ χ(G_1) ·χ{G_2) · χ{G_3). It is shown for positive integers a, b, c, and n with n ≥ a ≥ b ≥ c and a + b + c - 3 ≤ n ≤ abc that there exists a factorization {G_1, G_2, G_3} of K_n such that χ(G_1)=a, χ(G_2) = b, and χ(G_3) = c if and only if either (A) c = 1 and a + b - 1 ≤ n ≤ ab or (B) c ≥ 2.