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 ⌈ 2 n ⌉ ≤ 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.