The Art of Color Theory in Graphs: Coloring Algorithms and Their Diverse Applications
DOI:
https://doi.org/10.65419/albahit.v5i4.165Keywords:
graph coloring, chromatic number, DSATUR, greedy algorithms, integer programming, tabu search, reproducibility, schedulingAbstract
Graph vertex coloring assigns labels to vertices so that adjacent vertices receive different labels; its optimization objective is to minimize the number of labels. This manuscript develops a mathematically grounded review of the problem, its algorithmic families, and its applications in timetabling, frequency assignment, register allocation, and resource scheduling. It defines the principal graph parameters and proves elementary bounds, corrects the hypotheses and interpretation of Brooks’ theorem, and distinguishes the decision problem from the optimization problem. It describes fixed-order greedy and Welsh–Powell procedures, DSATUR, exact integer programming and branch-and-bound approaches, tabu search, and evolutionary methods. Complexity statements are qualified by data structure and implementation assumptions. The source manuscript reported numerical comparisons for graphs described as having 500 vertices and density 0.5, alongside broader runtime and optimality claims, but supplied no graph files, random seeds, code, hardware description, trial-level outcomes, or certified optimum/lower bounds. Those numbers therefore cannot be independently validated and are not retained as empirical findings. Instead, the paper provides a complete evaluation protocol using public benchmark provenance, independent runs, coloring validation, bound-aware gaps, and transparent reporting. The verified contribution is a coherent theoretical synthesis and a reproducible design for future empirical comparison; it does not establish that any one heuristic is universally superior
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