On $\omega \psi$-Perfect Graphs

TL;DR

This paper characterizes graphs where the clique number equals the pseudoachromatic number for all induced subgraphs, extending the concept of perfect graphs to new parameters related to graph coloring.

Contribution

It provides a characterization of b-perfect graphs specifically when a= and b=, generalizing perfect graph theory to new coloring parameters.

Findings

Characterization of -perfect graphs.

Extension of perfect graph concept to  and  parameters.

Generalization of graph perfection to coloring-related parameters.

Abstract

In this paper, we generalize the concept of {\it{perfect graphs}} to other parameters related to graph vertex coloring. This idea was introduced by Christen and Selkow in 1979 and Yegnanarayanan in 2001. Let $ a,b \in \{ \omega, \chi, \Gamma, \alpha, \psi \} $ where $ \omega $ is the clique number, $ \chi $ is the chromatic number, $ \Gamma $ is the Grundy number, $ \alpha $ is the achromatic number and $ \psi $ is the pseudoachromatic number. A graph $ G $ is \emph{$ ab $-perfect}, if for every induced subgraph $ H $ of $G$, $ a(H)$ equals $b(H) $. In this paper, we characterize the $ab$-perfect graphs when $a=\omega$ and $b=\psi$.