The graph Ramsey number R(G,H) is the smallest integer r such that every 2-coloring of the edges of Kr contains either a red copy of G or a blue copy of H . We find the largest star that can be removed from Kr such that the underlying graph is still forced to have a red G or a blue H . Thus, we introduce the star-critical Ramsey number r∗(G,H) as the smallest integer k such that every 2-colorin...

Synthesis of zeolite from kaolinite clay is a very promising process, requiring thorough beneficiation and calcinations to obtain purer monomer. Kaolinite clay sourced from Kankara, Nigeria is associated with inherent impurities, making its transformation rather challenging. The addition of dye, though, assisted in inhibiting the negative effects of impure phases on the desired end products and...

The aim of this study is to evaluate vital dyes "Brilliant Blue G" (BBG) and "Membrane Blue Dual" (MBD) for intraoperative staining of the inner limiting membrane (ILM) during vitrectomy for macular hole (MH). Retrospective, comparative case series on 18 eyes with macular holes who underwent "23 and 25 gauge" pars plana vitrectomy. Main outcome measurements were staining intensity and character...

Let G and H be simple graphs. The Ramsey number r(G,H) for a pair of graphs G and H is the smallest number r such that any red-blue coloring of the edges of Kr contains a red subgraph G or a blue subgraph H . The size Ramsey number r̂(G,H) for a pair of graphs G and H is the smallest number r̂ such that there exists a graph F with size r̂ satisfying the property that any red-blue coloring of the e...

In a red-blue coloring of a nonempty graph, every edge is colored red or blue. If the resulting edge-colored graph contains a nonempty subgraph G without isolated vertices every edge of which is colored the same, then G is said to be monochromatic. For two nonempty graphs G and H without isolated vertices, the monochromatic Ramsey number mr(G,H) of G and H is the minimum integer n such that eve...

For graphs G and H, the size balanced multipartite Ramsey number ) , ( H G m j is defined as the smallest positive integer s such that any arbitrary two red/blue coloring of the graph s j K × forces the appearance of a red G or a blue H . In this paper we find the exact values of the multipartite Ramsey numbers ) , ( 2 3 nK P m j and ) , ( 2 4 nK P m j .

Let G be an edge-bicolored graph where each edge is colored either red or blue. We study problems of obtaining an induced subgraph H from G that simultaneously satisfies given properties for H ’s red graph and blue graph. In particular, we considerDually Connected Induced Subgraph problem — find from G a k-vertex induced subgraph whose red and blue graphs are both connected, and Dual Separator ...

The graph Ramsey number R(G,H) is the smallest integer n such that every 2-coloring of the edges of Kn contains either a red copy of G or a blue copy of H . We find the largest star that can be removed from Kn such that the underlying graph is still forced to have a red G or a blue H . Thus, we introduce the star-avoiding Ramsey number r∗(G,H) as the smallest integer k such that every 2-colorin...

Two players, Red and Blue, each independently choose a vertex of a connected graph G. Red must then pay Blue an amount equal to the distance between the vertices chosen. In this note, w e investigate the value v(G) of this pursuit-evasion game for various classes of graphs G, as well as those optimal mixed strategies for achieving v(G). It is shown that some rather counterintuitive behavior can...

In the Red-Blue Dominating Set problem, we are given a bipartite graph G = (VB ∪ VR, E) and an integer k, and asked whether G has a subset D ⊆ VB of at most k ‘blue’ vertices such that each ‘red’ vertex from VR is adjacent to a vertex in D. We provide the first explicit linear kernel for this problem on planar graphs.