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 + ɛ ) ‐precoloring of vertices at pairwise distances at least d of a fractionally k ‐colorable graph can be extended to a fractional ( k + ɛ ) ‐coloring of the whole graph? The exact values of ε were known for k ∈ { 2 } ∪ [ 3 , ∞ ) and any d . We determine the exact values of ε for k ∈ ( 2 , 3 ) if d = 4 , and k ∈ [ 2.5 , 3 ) if d = 6 , and give upper bounds for k ∈ ( 2 , 3 ) if d = 5 , 7 , and k ∈ ( 2 , 2.5 ) if d = 6 . Surprisingly, ε viewed as a function of k is discontinuous for all those values of  d .