Given a graph G, a k-sparse j-set is a set of j vertices inducing a subgraph with maximum
degree at most k. A k-dense i-set is a set of i vertices that is k-sparse in the complement of G …

Let t be a nonnegative integer and G=(V (G), E (G)) be a graph. For v∈ V (G), let NG (v)={u∈
V (G)\{v}: uv∈ E (G)}. And for S⊆ V (G), we define d S (G; v)=| NG (v)∩ S| for v∈ S and d S …

In this paper, we initiate the study of defective Ramsey numbers for the class of perfect
graphs. Let PG be the class of all perfect graphs and R 1 PG (i, j) denote the smallest n such …

A set of vertices is $ k $-sparse if it induces a graph with a maximum degree of at most $ k $.
In this missive, we consider the order of the largest $ k $-sparse set in a triangle-free graph …

We consider the generalization of Ramsey numbers to the defective framework using k-
dense and k-sparse sets. We construct the first tableaux for defective Ramsey numbers with …

In this paper, we investigate a variant of Ramsey numbers called defective Ramsey numbers
where cliques and independent sets are generalized to k-dense and k-sparse sets, both …

We consider, for a positive integer $ k $, induced subgraphs in which each component has
order at most $ k $. Such a subgraph is said to be $ k $-divided. We show that finding large …

Given a graph $ G $, a $ k $-sparse $ j $-set is a set of $ j $ vertices inducing a subgraph
with maximum degree at most $ k $. A $ k $-dense $ i $-set is a set of $ i $ vertices that is $ k …

We investigate a variant of Ramsey numbers called defective Ramsey numbers, introduced
by Ekim and Gimbel in 2013, where cliques and independent sets are generalized to k …

One of the central notions in graph theory is that of a coloring–a partition of the vertices
where each part induces a graph with some given property. The most studied property is that …