Probing Graph Proper Total Colorings With Additional Constrained Conditions

TL;DR

This paper introduces new types of graph proper total colorings with additional constraints, explores their chromatic numbers, and verifies their existence on certain graphs, advancing theoretical understanding in graph coloring.

Contribution

It defines novel constrained total colorings and investigates their chromatic numbers, providing new insights and approximation methods in graph coloring theory.

Findings

New constrained total colorings defined

Chromatic number approximations developed

Existence verified on specific graphs

Abstract

Graph colorings are becoming an increasingly useful family of mathematical models for a broad range of applications, such as time tabling and scheduling, frequency assignment, register allocation, computer security and so on. Graph proper total colorings with additional constrained conditions have been investigated intensively in the last decade year. In this article some new graph proper total colorings with additional constrained conditions are defined, and approximations to the chromatic numbers of these colorings are researched, as well as some graphs having these colorings have been verified.