On Criticality and Additivity of the Pseudoachromatic Number Under Join

TL;DR

This paper investigates the properties of the pseudoachromatic number in graph theory, focusing on its additivity under graph join operations, correcting previous errors, and introducing the concept of weakly critical graphs.

Contribution

It provides new insights into the additivity of the pseudoachromatic number, corrects existing literature errors, and introduces the notion of weakly critical graphs for this study.

Findings

Pseudoachromatic number's additivity properties are characterized.

Errors in previous literature are identified and corrected.

The concept of weakly critical graphs is introduced for analysis.

Abstract

A vertex coloring of a graph is said to be pseudocomplete if, for any two distinct colors, there exists at least one edge with those two colors as its end vertices. The pseudoachromatic number of a graph is the greatest number of colors possible used in a pseudocomplete coloring. This paper studies properties relating to additivity of the pseudoachromatic number under the join. Errors from the literature are corrected and the notion of weakly critical is introduced in order to study the problem.