6 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
Changing upper irredundance by edge addition
AbstractDenote the upper irredundance number of a graph G by IR(G). A graph G is IR-edge-addition-sensitive if its upper irredundance number changes whenever an edge of Ḡ is added to G. Specifically, G is IR-edge-critical (IR+-edge-critical, respectively) if IR(G+e)<IR(G) (IR(G+e)>IR(G), respectively) for each edge e of Ḡ. We show that if G is IR-edge-addition-sensitive, then G is either IR-edge-critical or IR+-edge-critical. We obtain properties of the latter class of graphs, particularly in the case where β(G)=IR(G)=2 (where β(G) denotes the vertex independence number of G). This leads to an infinite class of IR+-edge-critical graphs where IR(G)=2
Irredundant and Mixed Ramsey Numbers
The irredundant Ramsey number, s(m,n), is the smallest p such that in every two-coloring of the edges of K[subscript]p using colors red (R) and blue (B), either the blue subgraph contains an m-element irredundant set or the red subgraph contains an n-element irredundant set. The mixed irredundant Ramsey number, t(m,n), is the smallest number p such that in every two-coloring of the edges of K[subscript]p using colors red (R) and blue (B), either the blue subgraph contains an m-element irredundant set or the red subgraph contains an n-element independent set. This thesis provides all known results for irredundant and mixed Ramsey numbers.  M.A