Conflict-Free Coloring of Intersection Graphs
A conflict-free k-coloring of a graph G=(V,E) assigns one of k different colors to some of the vertices such that,
for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors.
Such colorings have applications in wireless networking, robotics, and geometry, and are well studied in graph theory.
Here we study the conflict-free coloring of geometric intersection graphs.
We demonstrate that the intersection graph of n geometric objects without fatness properties and size restrictions may have conflict-free chromatic number in \Omega(log n/log log n) and in \Omega(\sqrt{\log n}) for disks or squares of different sizes;
it is known for general graphs that the worst case is in \Theta(log^2 n).
For unit-disk intersection graphs, we prove that it is NP-complete
to decide the existence of a conflict-free coloring
with one color; we also show that six colors always suffice,
using an algorithm that colors unit disk graphs of restricted height with two colors.
We conjecture that four colors are sufficient, which we prove for unit squares instead of unit disks.
For interval graphs, we establish a tight worst-case bound of two.
conflict-free coloring
intersection graphs
unit disk graphs
complexity
worst-case bounds
31:1-31:12
Regular Paper
Sándor P.
Fekete
Sándor P. Fekete
Phillip
Keldenich
Phillip Keldenich
10.4230/LIPIcs.ISAAC.2017.31
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