Nordhaus–Gaddum inequalities for the fractional and circular chromatic numbers

For a graph GG on nn vertices with chromatic number χ(G)χ(G), the Nordhaus–Gaddum inequalities state that ⌈2n⌉≤χ(G)+χ(G¯)≤n+1, and n≤χ(G)⋅χ(G¯)≤⌊(n+12)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.