Compelling Colorings: A generalization of the dominator chromatic number

TL;DR

This paper introduces a generalized graph coloring concept called $P$-compelling coloring, unifying various chromatic numbers and exploring bounds and algorithms for specific properties of vertex subsets.

Contribution

It defines the $P$-compelling chromatic number, generalizes existing chromatic concepts, and provides bounds and algorithms for particular properties of vertex subsets.

Findings

Generalizes dominator and total dominator chromatic numbers

Provides bounds for $P$-compelling chromatic number

Develops algorithms for specific cases

Abstract

We define a $P$-compelling coloring as a proper coloring of the vertices of a graph such that every subset consisting of one vertex of each color has property $P$. The $P$-compelling chromatic number is the minimum number of colors in such a coloring. We show that this notion generalizes the dominator and total dominator chromatic numbers, and provide some general bounds and algorithmic results. We also investigate the specific cases where $P$ is that the subset contains at least one edge or that the subset is connected.