93 research outputs found
Exploiting -Closure in Kernelization Algorithms for Graph Problems
A graph is c-closed if every pair of vertices with at least c common
neighbors is adjacent. The c-closure of a graph G is the smallest number such
that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated
it in the context of clique enumeration. We show that c-closure can be applied
in kernelization algorithms for several classic graph problems. We show that
Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a
kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with
O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed
graphs have polynomially-bounded Ramsey numbers, as we show
Irredundant sets, Ramsey numbers, multicolor Ramsey numbers
A set of vertices in a simple graph is irredundant if
each vertex is either isolated in the induced subgraph or else
has a private neighbor that is adjacent to and to no
other vertex of . The \emph{mixed Ramsey number} is the smallest
for which every red-blue coloring of the edges of has an -element
irredundant set in a blue subgraph or a -element independent set in a red
subgraph. The \emph{multicolor irredundant Ramsey number}
is the minimum such that every -coloring of the
edges of the complete graph on vertices has a monochromatic
irredundant set of size for certain .
Firstly, we improve the upper bound for the mixed Ramsey number , and
using this result, we verify a special case of a conjecture proposed by Chen,
Hattingh, and Rousseau for . Secondly, we obtain a new upper bound for
, and using Krivelevich's method, we establish an asymptotic lower
bound for CO-irredundant Ramsey number of , which extends Krivelevich's
result on . Thirdly, we prove a lower bound for the multicolor
irredundant Ramsey number by a random and probability method which has been
used to improve the lower bound for multicolor Ramsey numbers. Finally, we give
a lower bound for the irredundant multiplicity.Comment: 23 pages, 1 figur
Ramsey-type results on parameters related to domination
There is a philosophy to discover Ramsey-type theorem: given a graph
parameter , characterize the family \HH of graphs which satisfies that
every \HH-free graph has bounded parameter . The classical Ramsey's
theorem deals the parameter as the number of vertices. It also has a
corresponding connected version. This Ramsey-type problem on domination number
has been solved by Furuya. We will use this result to handle more parameters
related to domination.Comment: 12 pages, 1 figures
A Note on sparse supersaturation and extremal results for linear homogeneous systems
We study the thresholds for the property of containing a solution to a linear homogeneous system in random sets. We expand a previous sparse Sz\'emeredi-type result of Schacht to the broadest class of matrices possible. We also provide a shorter proof of a sparse Rado result of Friedgut, R\Postprint (published version
- …