Extensions of fractional precolorings show discontinuous behavior

We study the following problem: given a real number k and an integer d, what is the smallest ε such that any fractional ( k + ε ) -precoloring of vertices at pairwise distances at least d of a fractionally k-colorable graph can be extended to a fractional ( k + ε ) -coloring of the whole graph? The exact values of ε were known for k ϵ {2} ᴜ [3,∞] and any d. We determine the exact values of ε for k ϵ (2,3) if d = 4, and k ϵ [2.5,3) if d = 6, and give upper bounds for k ϵ (2,3) if d = 5,7, and k ϵ (2,2.5) if d = 6. Surprisingly, ε viewed as a function of k is discontinuous for all those values of d.