11,026 research outputs found
Bandwidth, expansion, treewidth, separators, and universality for bounded degree graphs
We establish relations between the bandwidth and the treewidth of bounded
degree graphs G, and relate these parameters to the size of a separator of G as
well as the size of an expanding subgraph of G. Our results imply that if one
of these parameters is sublinear in the number of vertices of G then so are all
the others. This implies for example that graphs of fixed genus have sublinear
bandwidth or, more generally, a corresponding result for graphs with any fixed
forbidden minor. As a consequence we establish a simple criterion for
universality for such classes of graphs and show for example that for each
gamma>0 every n-vertex graph with minimum degree ((3/4)+gamma)n contains a copy
of every bounded-degree planar graph on n vertices if n is sufficiently large
Labeling Schemes for Bounded Degree Graphs
We investigate adjacency labeling schemes for graphs of bounded degree
. In particular, we present an optimal (up to an additive
constant) adjacency labeling scheme for bounded degree trees.
The latter scheme is derived from a labeling scheme for bounded degree
outerplanar graphs. Our results complement a similar bound recently obtained
for bounded depth trees [Fraigniaud and Korman, SODA 10], and may provide new
insights for closing the long standing gap for adjacency in trees [Alstrup and
Rauhe, FOCS 02]. We also provide improved labeling schemes for bounded degree
planar graphs. Finally, we use combinatorial number systems and present an
improved adjacency labeling schemes for graphs of bounded degree with
Universal targets for homomorphisms of edge-colored graphs
A -edge-colored graph is a finite, simple graph with edges labeled by
numbers . A function from the vertex set of one -edge-colored
graph to another is a homomorphism if the endpoints of any edge are mapped to
two different vertices connected by an edge of the same color. Given a class
of graphs, a -edge-colored graph (not necessarily
with the underlying graph in ) is -universal for
when any -edge-colored graph with the underlying graph in
admits a homomorphism to . We characterize graph classes that admit
-universal graphs. For such classes, we establish asymptotically almost
tight bounds on the size of the smallest universal graph.
For a nonempty graph , the density of is the maximum ratio of the
number of edges to the number of vertices ranging over all nonempty subgraphs
of . For a nonempty class of graphs, denotes
the density of , that is the supremum of densities of graphs in
.
The main results are the following. The class admits
-universal graphs for if and only if there is an absolute constant
that bounds the acyclic chromatic number of any graph in . For any
such class, there exists a constant , such that for any , the size
of the smallest -universal graph is between and
.
A connection between the acyclic coloring and the existence of universal
graphs was first observed by Alon and Marshall (Journal of Algebraic
Combinatorics, 8(1):5-13, 1998). One of their results is that for planar
graphs, the size of the smallest -universal graph is between and
. Our results yield that there exists a constant such that for all
, this size is bounded from above by
Near-Optimal Induced Universal Graphs for Bounded Degree Graphs
A graph is an induced universal graph for a family of graphs if every
graph in is a vertex-induced subgraph of . For the family of all
undirected graphs on vertices Alstrup, Kaplan, Thorup, and Zwick [STOC
2015] give an induced universal graph with vertices,
matching a lower bound by Moon [Proc. Glasgow Math. Assoc. 1965].
Let . Improving asymptotically on previous results by
Butler [Graphs and Combinatorics 2009] and Esperet, Arnaud and Ochem [IPL
2008], we give an induced universal graph with vertices for the family of graphs with vertices of maximum degree
. For constant , Butler gives a lower bound of
. For an odd constant , Esperet et al.
and Alon and Capalbo [SODA 2008] give a graph with
vertices. Using their techniques for any
(including constant) even values of gives asymptotically worse bounds than
we present.
For large , i.e. when , the previous best
upper bound was due to Adjiashvili and
Rotbart [ICALP 2014]. We give upper and lower bounds showing that the size is
. Hence the optimal size is
and our construction is within a factor of
from this. The previous results were
larger by at least a factor of .
As a part of the above, proving a conjecture by Esperet et al., we construct
an induced universal graph with vertices for the family of graphs with
max degree . In addition, we give results for acyclic graphs with max degree
and cycle graphs. Our results imply the first labeling schemes that for any
are at most bits from optimal
Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity
We describe a linear-time algorithm that finds a planar drawing of every
graph of a simple line or pseudoline arrangement within a grid of area
O(n^{7/6}). No known input causes our algorithm to use area
\Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would
represent significant progress on the famous k-set problem from discrete
geometry. Drawing line arrangement graphs is the main task in the Planarity
puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing,
Bordeaux, 201
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