Extensions of Fractional Precolorings show Discontinuous Behavior

Jan van den Heuvel (1) , Daniel Král' (2) , Martin Kupec (3) , Jean-Sébastien Sereni (4) , Jan Volec (2)

1 LSE - Department Mathematics [London]
2 DIMAP - Centre for Discrete Mathematics and its Applications [Warwick]
3 ITI - Institut teoretické informatiky
4 ORPAILLEUR - Knowledge representation, reasonning

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∈{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.

Mots clés: fractional coloring; pre-coloring; extension

Domaines: Combinatoire [math.CO]

Jan van den Heuvel, Daniel Král', Martin Kupec, Jean-Sébastien Sereni, Jan Volec. Extensions of Fractional Precolorings show Discontinuous Behavior. [Research Report] Loria & Inria Grand Est. 2012. ⟨hal-00932924⟩