Extensions of Fractional Precolorings show Discontinuous Behavior

TL;DR

This paper investigates the extension of fractional precolorings in graphs, revealing that the minimal epsilon needed for extension exhibits discontinuous behavior across various parameters.

Contribution

It determines exact epsilon values for specific ranges of k and d, and uncovers the surprising discontinuity of epsilon as a function of k.

Findings

Exact epsilon values for k in (2,3) when d=4 and for k in [2.5,3) when d=6.

Upper bounds for epsilon when k in (2,3) for d=5,7 and when k in (2,2.5) for d=6.

Discontinuity of epsilon as a function of k across studied parameters.

Abstract

We study the following problem: given a real number k and integer d, what is the smallest epsilon such that any fractional (k+epsilon)-precoloring of vertices at pairwise distances at least d of a fractionally k-colorable graph can be extended to a fractional (k+epsilon)-coloring of the whole graph? The exact values of epsilon were known for k=2 and k\ge3 and any d. We determine the exact values of epsilon for k \in (2,3) if d=4, and k \in [2.5,3) if d=6, and give upper bounds for k \in (2,3) if d=5,7, and k \in (2,2.5) if d=6. Surprisingly, epsilon viewed as a function of k is discontinuous for all those values of d.