# Masarykova univerzita

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# Extensions of Fractional Precolorings Show Discontinuous Behavior

Autoři HEUVEL van den KUPEC M SERENI JS VOLEC J 2014 Článek v odborném periodiku Journal of Graph Theory http://dx.doi.org/10.1002/jgt.21787 fractional coloring; precoloring extension We study the following problem: given a real number k and an integer d, what is the smallest e 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 is an element of {2} boolean OR [3,infinity) and any d. We determine the exact values of epsilon for k is an element of (2, 3) if d = 4, and k is an element of [2.5, 3) if d = 6, and give upper bounds for k is an element of (2, 3) if d = 5, 7, and k is an element of (2, 2.5) if d = 6. Surprisingly, epsilon viewed as a function of k is discontinuous for all those values of d. (C) 2014 Wiley Periodicals, Inc.

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