11 research outputs found
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\u27s 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 of Planar Graphs
A conflict-free k-coloring of a graph 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 natural problem of the
conflict-free chromatic number chi_CF(G) (the smallest k for which
conflict-free k-colorings exist). We provide results both for closed
neighborhoods N[v], for which a vertex v is a member of its neighborhood, and
for open neighborhoods N(v), for which vertex v is not a member of its
neighborhood.
For closed neighborhoods, we prove the conflict-free variant of the famous
Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a
minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case
bound: three colors are sometimes necessary and always sufficient. We also give
a complete characterization of the computational complexity of conflict-free
coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G,
but polynomial for outerplanar graphs. Furthermore, deciding whether
chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for
outerplanar graphs. For the bicriteria problem of minimizing the number of
colored vertices subject to a given bound k on the number of colors, we give a
full algorithmic characterization in terms of complexity and approximation for
outerplanar and planar graphs.
For open neighborhoods, we show that every planar bipartite graph has a
conflict-free coloring with at most four colors; on the other hand, we prove
that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite
graph has a conflict-free k-coloring. Moreover, we establish that any general}
planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on
Discrete Mathematics) of extended abstract that appears in Proceeedings of
the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
2017), pp. 1951-196
Extremal Results on Conflict-free Coloring
A conflict-free open neighborhood coloring of a graph is an assignment of
colors to the vertices such that for every vertex there is a color that appears
exactly once in its open neighborhood. For a graph , the smallest number of
colors required for such a coloring is called the conflict-free open
neighborhood (CFON) chromatic number and is denoted by . By
considering closed neighborhood instead of open neighborhood, we obtain the
analogous notions of conflict-free closed neighborhood (CFCN) coloring, and
CFCN chromatic number (denoted by ). The notion of conflict-free
coloring was introduced in 2002, and has since received considerable attention.
In this paper, we study some extremal questions related to CFON and CFCN
coloring.Comment: 18 page
The potential to improve the choice: list conflict-free coloring for geometric hypergraphs
Given a geometric hypergraph (or a range-space) , a coloring of its vertices is said to be conflict-free if for every hyperedge there is at least one vertex in whose color is distinct from the colors of all other vertices in . The study of this notion is motivated by frequency assignment problems in wireless networks. We study the list-coloring (or choice) version of this notion. In this version, each vertex is associated with a set of (admissible) colors and it is allowed to be colored only with colors from its set. List coloring arises naturally in the context of wireless networks. Our main result is a list coloring algorithm based on a new potential method. The algorithm produces a stronger unique-maximum coloring, in which colors are positive integers and the maximum color in every hyperedge occurs uniquely. As a corollary, we provide asymptotically sharp bounds on the size of the lists required to assure the existence of such unique-maximum colorings for many geometric hypergraphs (e.g., discs or pseudo-discs in the plane or points with respect to discs). Moreover, we provide an algorithm, such that, given a family of lists with the appropriate sizes, computes such a coloring from these lists