Dey, Jatin Kumar2025-02-052025-02-052024-0642p.http://hdl.handle.net/10263/7503Dissertation under the supervision of Dr. Mathew C. Francis.This thesis explores the application of the entropy compression technique to various graph coloring problems, offering an innovative approach to addressing significant challenges in graph theory. Entropy compression, particularly the Moser-Tardos [12] framework, transforms probabilistic existence proofs into explicit, constructive algorithms, thereby enhancing our understanding and expanding the toolkit available for solving these challenges. Graph coloring problems involve assigning colors to the vertices or edges of a graph under specific constraints, such as ensuring no two adjacent vertices or edges share the same color. These problems are both theoretically rich and practically significant, with applications in scheduling, register allocation in compilers, and network frequency assignment. Motivated by the work of Esperet and Parreau [6], this research focuses on the acyclic edge chromatic number. Through rigorous analysis and algorithmic design, this study demonstrates how entropy compression has the potential to improve existing bounds to complex combinatorial problems.enAcyclic Edge Chromatic NumberDiagonal Ramsey Numberk-Uniform HypergraphOther