Extensions of Fractional Precolorings Show Discontinuous Behavior

Abstract

We study the following problem: given a real number k and 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 ∈ {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.