50 research outputs found
Coloring d-Embeddable k-Uniform Hypergraphs
This paper extends the scenario of the Four Color Theorem in the following
way. Let H(d,k) be the set of all k-uniform hypergraphs that can be (linearly)
embedded into R^d. We investigate lower and upper bounds on the maximum (weak
and strong) chromatic number of hypergraphs in H(d,k). For example, we can
prove that for d>2 there are hypergraphs in H(2d-3,d) on n vertices whose weak
chromatic number is Omega(log n/log log n), whereas the weak chromatic number
for n-vertex hypergraphs in H(d,d) is bounded by O(n^((d-2)/(d-1))) for d>2.Comment: 18 page
Unsplittable coverings in the plane
A system of sets forms an {\em -fold covering} of a set if every point
of belongs to at least of its members. A -fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body , there exists a constant such that every
-fold covering of the plane with translates of splits into
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every , an
unsplittable -fold covering of the plane with translates of any open convex
body which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets do not misbehave,
they satisfy the conjecture: every -fold covering of any region of the plane
by translates of such a set splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant such that, for
any positive integer , every -fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} sets
Chromatic numbers of Cayley graphs of abelian groups: A matrix method
In this paper, we take a modest first step towards a systematic study of
chromatic numbers of Cayley graphs on abelian groups. We lose little when we
consider these graphs only when they are connected and of finite degree. As in
the work of Heuberger and others, in such cases the graph can be represented by
an integer matrix, where we call the dimension and the
rank. Adding or subtracting rows produces a graph homomorphism to a graph with
a matrix of smaller dimension, thereby giving an upper bound on the chromatic
number of the original graph. In this article we develop the foundations of
this method. In a series of follow-up articles using this method, we completely
determine the chromatic number in cases with small dimension and rank; prove a
generalization of Zhu's theorem on the chromatic number of -valent integer
distance graphs; and provide an alternate proof of Payan's theorem that a
cube-like graph cannot have chromatic number 3.Comment: 17 page
Chromatic numbers of Cayley graphs of abelian groups: Cases of small dimension and rank
A connected Cayley graph on an abelian group with a finite generating set
can be represented by its Heuberger matrix, i.e., an integer matrix whose
columns generate the group of relations between members of . In a companion
article, the authors lay the foundation for the use of Heuberger matrices to
study chromatic numbers of abelian Cayley graphs. We call the number of rows in
the Heuberger matrix the dimension, and the number of columns the rank. In this
paper, we give precise numerical conditions that completely determine the
chromatic number in all cases with dimension ; with rank ; and with
dimension and rank . For such a graph without loops, we show
that it is -colorable if and only if it does not contain a -clique, and
it is -colorable if and only if it contains neither a diamond lanyard nor a
, both of which we define herein. In a separate companion article,
we show that we recover Zhu's theorem on the chromatic number of -valent
integer distance graphs as a special case of our theorem for dimension and
rank .Comment: 27 page
Recognizing Geometric Intersection Graphs Stabbed by a Line
In this paper, we determine the computational complexity of recognizing two
graph classes, \emph{grounded L}-graphs and \emph{stabbable grid intersection}
graphs. An L-shape is made by joining the bottom end-point of a vertical
() segment to the left end-point of a horizontal () segment. The top
end-point of the vertical segment is known as the {\em anchor} of the L-shape.
Grounded L-graphs are the intersection graphs of L-shapes such that all the
L-shapes' anchors lie on the same horizontal line. We show that recognizing
grounded L-graphs is NP-complete. This answers an open question asked by
Jel{\'\i}nek \& T{\"o}pfer (Electron. J. Comb., 2019).
Grid intersection graphs are the intersection graphs of axis-parallel line
segments in which two vertical (similarly, two horizontal) segments cannot
intersect. We say that a (not necessarily axis-parallel) straight line
stabs a segment , if intersects . A graph is a stabbable grid
intersection graph () if there is a grid intersection representation
of in which the same line stabs all its segments. We show that recognizing
graphs is -complete, even on a restricted class of graphs. This
answers an open question asked by Chaplick \etal (\textsc{O}rder, 2018).Comment: 18 pages, 11 Figure