Defective Ramsey Numbers and Defective Cocolorings in Some Subclasses of Perfect Graphs

TL;DR

This paper explores defective Ramsey numbers within subclasses of perfect graphs, introduces an algorithm for their computation, and provides new numerical results along with studying related partition parameters.

Contribution

It presents a generic algorithm for computing defective Ramsey numbers in specific graph classes and reports new computed values for perfect graphs, bipartite, and chordal graphs.

Findings

Computed new defective Ramsey numbers for perfect graphs, bipartite graphs, and chordal graphs.

Introduced and analyzed the parameter c^{ ext{G}}_k(m) for graph partitions.

Provided an efficient graph generation algorithm for these computations.

Abstract

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 commonly called $k$-defective sets. We focus on the computation of defective Ramsey numbers restricted to some subclasses of perfect graphs. Since direct proof techniques are often insufficient for obtaining new values of defective Ramsey numbers, we provide a generic algorithm to compute defective Ramsey numbers in a given target graph class. We combine direct proof techniques with our efficient graph generation algorithm to compute several new defective Ramsey numbers in perfect graphs, bipartite graphs and chordal graphs. We also initiate the study of a related parameter, denoted by $c^{\mathcal G}_k(m)$, which is the maximum order $n$ such that the vertex set of any graph of order at $n$ in a class $\mathcal{G}$ can be partitioned into at most $m$ subsets each of which is $k$-defective. We obtain several values for $c^{\mathcal G}_k(m)$ in perfect graphs and cographs.