10,039 research outputs found
Short proofs of some extremal results
We prove several results from different areas of extremal combinatorics,
giving complete or partial solutions to a number of open problems. These
results, coming from areas such as extremal graph theory, Ramsey theory and
additive combinatorics, have been collected together because in each case the
relevant proofs are quite short.Comment: 19 page
On globally sparse Ramsey graphs
We say that a graph has the Ramsey property w.r.t.\ some graph and
some integer , or is -Ramsey for short, if any -coloring
of the edges of contains a monochromatic copy of . R{\"o}dl and
Ruci{\'n}ski asked how globally sparse -Ramsey graphs can possibly
be, where the density of is measured by the subgraph with
the highest average degree. So far, this so-called Ramsey density is known only
for cliques and some trivial graphs . In this work we determine the Ramsey
density up to some small error terms for several cases when is a complete
bipartite graph, a cycle or a path, and colors are available
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
Dismantling sparse random graphs
We consider the number of vertices that must be removed from a graph G in
order that the remaining subgraph has no component with more than k vertices.
Our principal observation is that, if G is a sparse random graph or a random
regular graph on n vertices with n tending to infinity, then the number in
question is essentially the same for all values of k such that k tends to
infinity but k=o(n).Comment: 7 page
Slider-pinning Rigidity: a Maxwell-Laman-type Theorem
We define and study slider-pinning rigidity, giving a complete combinatorial
characterization. This is done via direction-slider networks, which are a
generalization of Whiteley's direction networks.Comment: Accepted, to appear in Discrete and Computational Geometr
Planar Induced Subgraphs of Sparse Graphs
We show that every graph has an induced pseudoforest of at least
vertices, an induced partial 2-tree of at least vertices, and an
induced planar subgraph of at least vertices. These results are
constructive, implying linear-time algorithms to find the respective induced
subgraphs. We also show that the size of the largest -minor-free graph in
a given graph can sometimes be at most .Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph
Algorithms and Application
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