16,178 research outputs found
Expanders Are Universal for the Class of All Spanning Trees
Given a class of graphs F, we say that a graph G is universal for F, or
F-universal, if every H in F is contained in G as a subgraph. The construction
of sparse universal graphs for various families F has received a considerable
amount of attention. One is particularly interested in tight F-universal
graphs, i.e., graphs whose number of vertices is equal to the largest number of
vertices in a graph from F. Arguably, the most studied case is that when F is
some class of trees.
Given integers n and \Delta, we denote by T(n,\Delta) the class of all
n-vertex trees with maximum degree at most \Delta. In this work, we show that
every n-vertex graph satisfying certain natural expansion properties is
T(n,\Delta)-universal or, in other words, contains every spanning tree of
maximum degree at most \Delta. Our methods also apply to the case when \Delta
is some function of n. The result has a few very interesting implications. Most
importantly, we obtain that the random graph G(n,p) is asymptotically almost
surely (a.a.s.) universal for the class of all bounded degree spanning (i.e.,
n-vertex) trees provided that p \geq c n^{-1/3} \log^2n where c > 0 is a
constant. Moreover, a corresponding result holds for the random regular graph
of degree pn. In fact, we show that if \Delta satisfies \log n \leq \Delta \leq
n^{1/3}, then the random graph G(n,p) with p \geq c \Delta n^{-1/3} \log n and
the random r-regular n-vertex graph with r \geq c\Delta n^{2/3} \log n are
a.a.s. T(n,\Delta)-universal. Another interesting consequence is the existence
of locally sparse n-vertex T(n,\Delta)-universal graphs. For constant \Delta,
we show that one can (randomly) construct n-vertex T(n,\Delta)-universal graphs
with clique number at most five. Finally, we show robustness of random graphs
with respect to being universal for T(n,\Delta) in the context of the
Maker-Breaker tree-universality game.Comment: 25 page
Extremal density for sparse minors and subdivisions
We prove an asymptotically tight bound on the extremal density guaranteeing
subdivisions of bounded-degree bipartite graphs with a mild separability
condition. As corollaries, we answer several questions of Reed and Wood on
embedding sparse minors. Among others,
average degree is sufficient to force the
grid as a topological minor;
average degree forces every -vertex planar graph
as a minor, and the constant is optimal, furthermore, surprisingly, the
value is the same for -vertex graphs embeddable on any fixed surface;
a universal bound of on average degree forcing every
-vertex graph in any nontrivial minor-closed family as a minor, and the
constant 2 is best possible by considering graphs with given treewidth.Comment: 33 pages, 6 figure
Extremal density for sparse minors and subdivisions
We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse minors. Among others,
∙ (1+o(1))t2 average degree is sufficient to force the t×t grid as a topological minor;
∙ (3/2+o(1))t average degree forces every t-vertex planar graph as a minor, and the constant 3/2 is optimal, furthermore, surprisingly, the value is the same for t-vertex graphs embeddable on any fixed surface;
∙ a universal bound of (2+o(1))t on average degree forcing every t-vertex graph in any nontrivial minor-closed family as a minor, and the constant 2 is best possible by considering graphs with given treewidth
Embedding nearly-spanning bounded degree trees
We derive a sufficient condition for a sparse graph G on n vertices to
contain a copy of a tree T of maximum degree at most d on (1-\epsilon)n
vertices, in terms of the expansion properties of G. As a result we show that
for fixed d\geq 2 and 0<\epsilon<1, there exists a constant c=c(d,\epsilon)
such that a random graph G(n,c/n) contains almost surely a copy of every tree T
on (1-\epsilon)n vertices with maximum degree at most d. We also prove that if
an (n,D,\lambda)-graph G (i.e., a D-regular graph on n vertices all of whose
eigenvalues, except the first one, are at most \lambda in their absolute
values) has large enough spectral gap D/\lambda as a function of d and
\epsilon, then G has a copy of every tree T as above
Near-optimal adjacency labeling scheme for power-law graphs
An adjacency labeling scheme is a method that assigns labels to the vertices
of a graph such that adjacency between vertices can be inferred directly from
the assigned label, without using a centralized data structure. We devise
adjacency labeling schemes for the family of power-law graphs. This family that
has been used to model many types of networks, e.g. the Internet AS-level
graph. Furthermore, we prove an almost matching lower bound for this family. We
also provide an asymptotically near- optimal labeling scheme for sparse graphs.
Finally, we validate the efficiency of our labeling scheme by an experimental
evaluation using both synthetic data and real-world networks of up to hundreds
of thousands of vertices
Sublinear Distance Labeling
A distance labeling scheme labels the nodes of a graph with binary
strings such that, given the labels of any two nodes, one can determine the
distance in the graph between the two nodes by looking only at the labels. A
-preserving distance labeling scheme only returns precise distances between
pairs of nodes that are at distance at least from each other. In this paper
we consider distance labeling schemes for the classical case of unweighted
graphs with both directed and undirected edges.
We present a bit -preserving distance labeling
scheme, improving the previous bound by Bollob\'as et. al. [SIAM J. Discrete
Math. 2005]. We also give an almost matching lower bound of
. With our -preserving distance labeling scheme as a
building block, we additionally achieve the following results:
1. We present the first distance labeling scheme of size for sparse
graphs (and hence bounded degree graphs). This addresses an open problem by
Gavoille et. al. [J. Algo. 2004], hereby separating the complexity from
distance labeling in general graphs which require bits, Moon [Proc.
of Glasgow Math. Association 1965].
2. For approximate -additive labeling schemes, that return distances
within an additive error of we show a scheme of size for .
This improves on the current best bound of by
Alstrup et. al. [SODA 2016] for sub-polynomial , and is a generalization of
a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for
.Comment: A preliminary version of this paper appeared at ESA'1
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