Nordhaus-Gaddum inequalities for the fractional and circular chromatic numbers

For a graph G on n vertices with chromatic number χ(G), the Nordhaus–Gaddum inequalities state that d2 √ ne ≤ χ(G) + χ(G) ≤ n + 1, and n ≤ χ(G) · χ(G) ≤ ⌊( 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.