872 research outputs found
Relaxed Strong Colorings of Hypergraphs
https://digitalcommons.morris.umn.edu/urs_2019/1004/thumbnail.jp
Conflict-Free Coloring Made Stronger
In FOCS 2002, Even et al. showed that any set of discs in the plane can
be Conflict-Free colored with a total of at most colors. That is,
it can be colored with colors such that for any (covered) point
there is some disc whose color is distinct from all other colors of discs
containing . They also showed that this bound is asymptotically tight. In
this paper we prove the following stronger results:
\begin{enumerate} \item [(i)] Any set of discs in the plane can be
colored with a total of at most colors such that (a) for any
point that is covered by at least discs, there are at least
distinct discs each of which is colored by a color distinct from all other
discs containing and (b) for any point covered by at most discs,
all discs covering are colored distinctively. We call such a coloring a
{\em -Strong Conflict-Free} coloring. We extend this result to pseudo-discs
and arbitrary regions with linear union-complexity.
\item [(ii)] More generally, for families of simple closed Jordan regions
with union-complexity bounded by , we prove that there exists
a -Strong Conflict-Free coloring with at most colors.
\item [(iii)] We prove that any set of axis-parallel rectangles can be
-Strong Conflict-Free colored with at most colors.
\item [(iv)] We provide a general framework for -Strong Conflict-Free
coloring arbitrary hypergraphs. This framework relates the notion of -Strong
Conflict-Free coloring and the recently studied notion of -colorful
coloring. \end{enumerate}
All of our proofs are constructive. That is, there exist polynomial time
algorithms for computing such colorings
Rainbow Coloring Hardness via Low Sensitivity Polymorphisms
A k-uniform hypergraph is said to be r-rainbow colorable if there is an r-coloring of its vertices such that every hyperedge intersects all r color classes. Given as input such a hypergraph, finding a r-rainbow coloring of it is NP-hard for all k >= 3 and r >= 2. Therefore, one settles for finding a rainbow coloring with fewer colors (which is an easier task). When r=k (the maximum possible value), i.e., the hypergraph is k-partite, one can efficiently 2-rainbow color the hypergraph, i.e., 2-color its vertices so that there are no monochromatic edges. In this work we consider the next smaller value of r=k-1, and prove that in this case it is NP-hard to rainbow color the hypergraph with q := ceil[(k-2)/2] colors. In particular, for k <=6, it is NP-hard to 2-color (k-1)-rainbow colorable k-uniform hypergraphs.
Our proof follows the algebraic approach to promise constraint satisfaction problems. It proceeds by characterizing the polymorphisms associated with the approximate rainbow coloring problem, which are rainbow colorings of some product hypergraphs on vertex set [r]^n. We prove that any such polymorphism f: [r]^n -> [q] must be C-fixing, i.e., there is a small subset S of C coordinates and a setting a in [q]^S such that fixing x_{|S} = a determines the value of f(x). The key step in our proof is bounding the sensitivity of certain rainbow colorings, thereby arguing that they must be juntas. Armed with the C-fixing characterization, our NP-hardness is obtained via a reduction from smooth Label Cover
-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum
A -WORM coloring of a graph is an assignment of colors to the
vertices in such a way that the vertices of each -subgraph of get
precisely two colors. We study graphs which admit at least one such
coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014)
161--173] who asked whether every such graph has a -WORM coloring with two
colors. In fact for every integer there exists a -WORM colorable
graph in which the minimum number of colors is exactly . There also exist
-WORM colorable graphs which have a -WORM coloring with two colors
and also with colors but no coloring with any of colors. We
also prove that it is NP-hard to determine the minimum number of colors and
NP-complete to decide -colorability for every (and remains
intractable even for graphs of maximum degree 9 if ). On the other hand,
we prove positive results for -degenerate graphs with small , also
including planar graphs. Moreover we point out a fundamental connection with
the theory of the colorings of mixed hypergraphs. We list many open problems at
the end.Comment: 18 page
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