55 research outputs found
Ramsey numbers of ordered graphs
An ordered graph is a pair where is a graph and
is a total ordering of its vertices. The ordered Ramsey number
is the minimum number such that every ordered
complete graph with vertices and with edges colored by two colors contains
a monochromatic copy of .
In contrast with the case of unordered graphs, we show that there are
arbitrarily large ordered matchings on vertices for which
is superpolynomial in . This implies that
ordered Ramsey numbers of the same graph can grow superpolynomially in the size
of the graph in one ordering and remain linear in another ordering.
We also prove that the ordered Ramsey number is
polynomial in the number of vertices of if the bandwidth of
is constant or if is an ordered graph of constant
degeneracy and constant interval chromatic number. The first result gives a
positive answer to a question of Conlon, Fox, Lee, and Sudakov.
For a few special classes of ordered paths, stars or matchings, we give
asymptotically tight bounds on their ordered Ramsey numbers. For so-called
monotone cycles we compute their ordered Ramsey numbers exactly. This result
implies exact formulas for geometric Ramsey numbers of cycles introduced by
K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of
Combinatoric
Classical Ising model test for quantum circuits
We exploit a recently constructed mapping between quantum circuits and graphs
in order to prove that circuits corresponding to certain planar graphs can be
efficiently simulated classically. The proof uses an expression for the Ising
model partition function in terms of quadratically signed weight enumerators
(QWGTs), which are polynomials that arise naturally in an expansion of quantum
circuits in terms of rotations involving Pauli matrices. We combine this
expression with a known efficient classical algorithm for the Ising partition
function of any planar graph in the absence of an external magnetic field, and
the Robertson-Seymour theorem from graph theory. We give as an example a set of
quantum circuits with a small number of non-nearest neighbor gates which admit
an efficient classical simulation.Comment: 17 pages, 2 figures. v2: main result strengthened by removing
oracular settin
Metric Embedding via Shortest Path Decompositions
We study the problem of embedding shortest-path metrics of weighted graphs
into spaces. We introduce a new embedding technique based on low-depth
decompositions of a graph via shortest paths. The notion of Shortest Path
Decomposition depth is inductively defined: A (weighed) path graph has shortest
path decomposition (SPD) depth . General graph has an SPD of depth if it
contains a shortest path whose deletion leads to a graph, each of whose
components has SPD depth at most . In this paper we give an
-distortion embedding for graphs of SPD
depth at most . This result is asymptotically tight for any fixed ,
while for it is tight up to second order terms.
As a corollary of this result, we show that graphs having pathwidth embed
into with distortion . For
, this improves over the best previous bound of Lee and Sidiropoulos that
was exponential in ; moreover, for other values of it gives the first
embeddings whose distortion is independent of the graph size . Furthermore,
we use the fact that planar graphs have SPD depth to give a new
proof that any planar graph embeds into with distortion . Our approach also gives new results for graphs with bounded treewidth,
and for graphs excluding a fixed minor
Graph Treewidth and Geometric Thickness Parameters
Consider a drawing of a graph in the plane such that crossing edges are
coloured differently. The minimum number of colours, taken over all drawings of
, is the classical graph parameter "thickness". By restricting the edges to
be straight, we obtain the "geometric thickness". By further restricting the
vertices to be in convex position, we obtain the "book thickness". This paper
studies the relationship between these parameters and treewidth.
Our first main result states that for graphs of treewidth , the maximum
thickness and the maximum geometric thickness both equal .
This says that the lower bound for thickness can be matched by an upper bound,
even in the more restrictive geometric setting. Our second main result states
that for graphs of treewidth , the maximum book thickness equals if and equals if . This refutes a conjecture of Ganley and
Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved
for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of
the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in
Computer Science 3843:129-140, Springer, 2006. The full version was published
in Discrete & Computational Geometry 37(4):641-670, 2007. That version
contained a false conjecture, which is corrected on page 26 of this versio
Packing chromatic vertex-critical graphs
The packing chromatic number of a graph is the smallest
integer such that the vertex set of can be partitioned into sets ,
, where vertices in are pairwise at distance at least .
Packing chromatic vertex-critical graphs, -critical for short, are
introduced as the graphs for which
holds for every vertex of . If , then is
--critical. It is shown that if is -critical,
then the set can be almost
arbitrary. The --critical graphs are characterized, and
--critical graphs are characterized in the case when they
contain a cycle of length at least which is not congruent to modulo
. It is shown that for every integer there exists a
--critical tree and that a --critical
caterpillar exists if and only if . Cartesian products are also
considered and in particular it is proved that if and are
vertex-transitive graphs and , then is -critical
On Upward Drawings of Trees on a Given Grid
Computing a minimum-area planar straight-line drawing of a graph is known to
be NP-hard for planar graphs, even when restricted to outerplanar graphs.
However, the complexity question is open for trees. Only a few hardness results
are known for straight-line drawings of trees under various restrictions such
as edge length or slope constraints. On the other hand, there exist
polynomial-time algorithms for computing minimum-width (resp., minimum-height)
upward drawings of trees, where the height (resp., width) is unbounded.
In this paper we take a major step in understanding the complexity of the
area minimization problem for strictly-upward drawings of trees, which is one
of the most common styles for drawing rooted trees. We prove that given a
rooted tree and a grid, it is NP-hard to decide whether
admits a strictly-upward (unordered) drawing in the given grid.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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