The bound for the number of colors which are sufficient for map coloring on a surface of genus g, γ(g) = ⌊1/2(7 + sqrt(48g + 1))⌋ is the best possible, where ⌊x⌋ is the floor function. γ(g) is called the chromatic number, and the first few values for g = 0, 1, ... are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, ... (OEIS A000934).
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