Extensions of Fractional Precolorings show Discontinuous Behavior
Combinatorics
2013-12-18 v2
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.
Keywords
Cite
@article{arxiv.1205.5405,
title = {Extensions of Fractional Precolorings show Discontinuous Behavior},
author = {Jan van den Heuvel and Daniel Kral and Martin Kupec and Jean-Sebastien Sereni and Jan Volec},
journal= {arXiv preprint arXiv:1205.5405},
year = {2013}
}