RAINBOW AND PROPERLY COLORED SUBGRAPHS

Graphs are foundational to combinatorics, modeling relationships across disciplines, from communication networks and social systems to molecular biology. This book presents the first comprehensive treatment of extremal problems in edge-colored graphs, a rapidly developing area at the intersection of graph theory, combinatorics, and applied mathematics.

Focusing on the existence of rainbow and properly colored subgraphs, the text explores classical and contemporary topics including anti-Ramsey theory, Gallai–Ramsey results, and rainbow Turán-type problems. It offers a unified framework for understanding how coloring constraints affect graph structure, while drawing connections to Latin squares, transversals, and real-world network models.

Rich in theory and application, this accessible and in-depth resource serves as both a reference for researchers and a guide for graduate students, bringing together decades of literature into a coherent and insightful narrative.

Contents:

• Preface
• Introduction
• Cycle
• Path, Matching, Tree and Clique
• Host Graphs with Certain Color Patterns
• Anti-Ramsey Theory
• Gallai-Ramsey Results and the Multicolor Erdős–Hajnal Conjecture
• Rainbow Turán Problem
• Uniformly Edge-Colored Graphs
• Transversal in Graph System
• Color-Bias Problem
• Bibliography

Readership: Graduate students and advanced undergraduates in mathematics and theoretical computer science, seeking a thorough and accessible introduction to edge-colored graph theory and its current research directions. Researchers and academics in graph theory, combinatorics, and discrete mathematics looking for a comprehensive reference that consolidates decades of developments in the field. Scientists and professionals in engineering, computer science, biology, physics, and chemistry who apply graph-theoretical methods to model communication networks, biological systems, and data structures.