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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 chromaticarXiv:1601.00883v1 fatcat:3tm4iea53bhadl7sbw6wkj3e2q