2 research outputs found
Strong Conflict-Free Coloring of Intervals
We consider the k-strong conflict-free coloring of a set of points on a line
with respect to a family of intervals: Each point on the line must be assigned
a color so that the coloring has to be conflict-free, in the sense that in
every interval I there are at least k colors each appearing exactly once in I.
In this paper, we present a polynomial algorithm for the general problem; the
algorithm has an approximation factor 5-2/k when k\geq2 and approximation
factor 2 for k=1. In the special case the family contains all the possible
intervals on the given set of points, we show that a 2 approximation algorithm
exists, for any k\geq1
On variants of conflict-free-coloring for hypergraphs
Conflict-free coloring is a kind of vertex coloring of hypergraphs requiring
each hyperedge to have a color which appears only on one vertex. More
generally, for a positive integer there are -conflict-free colorings
(-CF-colorings for short) and -strong-conflict-free colorings
(-SCF-colorings for short). %for some positive integer . Let be the
hypergraph of which the vertex-set is and the
hyperedge-set is the set of all (non-empty) subsets of
consisting of consecutive elements of . Firstly, we study the
-SCF-coloring of , give the exact -SCF-coloring chromatic number of
for , and present upper and lower bounds of the -SCF-coloring
chromatic number of for all . Secondly, we give the exact
-CF-coloring chromatic number of for all .Comment: 15 page