# Algorithmic complexity of proper labeling problems

## TL;DR

This paper investigates the computational complexity of various proper labeling problems in graphs, presenting polynomial algorithms for some cases and NP-completeness results for others.

## Contribution

It provides new insights into the algorithmic complexity of proper labeling variants, including polynomial algorithms and NP-completeness proofs.

## Key findings

- Polynomial time algorithms for certain proper labeling variants
- NP-completeness results for other labeling problems
- Enhanced understanding of the computational boundaries of proper labeling

## Abstract

A proper labeling of a graph is an assignment of integers to some elements of a graph, which may be the vertices, the edges, or both of them, such that we obtain a proper vertex coloring via the labeling subject to some conditions. The problem of proper labeling offers many variants and received a great interest during recent years. We consider the algorithmic complexity of some variants of the proper labeling problems, we present some polynomial time algorithms and $ \mathbf{NP} $-completeness results for them.