On Fair and Tolerant Colorings of Graphs
Saeed Shaebani

TL;DR
This paper investigates the concept of fair and tolerant vertex colorings in graphs, introducing the FAT chromatic number, and proves that certain bounds relating it to the traditional chromatic number do not exist.
Contribution
The paper establishes that no universal functions can bound the FAT chromatic number in terms of the traditional chromatic number, answering open questions negatively.
Findings
Both bounding functions do not exist for all graphs.
Negative resolution to open questions about FAT chromatic number.
Introduces and studies the FAT chromatic number concept.
Abstract
A (not necessarily proper) vertex coloring of a graph with color classes , , , , is said to be a {\it Fair And Tolerant vertex coloring of with colors}, whenever , , , are nonempty and there exist two real numbers and such that and and the following condition holds for each arbitrary vertex and every arbitrary color class : The {\it FAT chromatic number} of , denoted by , is defined as the maximum positive integer for which admits a Fair And Tolerant vertex coloring with colors. The concept of the FAT chromatic number of graphs was introduced and studied by Beers and Mulas,…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Advanced Topology and Set Theory
