3 research outputs found
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