Nordhaus-Gaddum inequalities for the fractional and circular chromatic numbers

Abstract

For a graph G on n vertices with chromatic number χ (G), the Nordhaus-Gaddum inequalities state that ⌈ 2 sqrt(n) ⌉ ≤ χ (G) + χ (over(G, -)) ≤ n + 1, and n ≤ χ (G) {dot operator} χ (over(G, -)) ≤ ⌊ (frac(n + 1, 2))2 ⌋. Much analysis has been done to derive similar inequalities for other graph parameters, all of which are integer-valued. We determine here the optimal Nordhaus-Gaddum inequalities for the circular chromatic number and the fractional chromatic number, the first examples of Nordhaus-Gaddum inequalities where the graph parameters are rational-valued. © 2008 Elsevier B.V. All rights reserved.