A Comprehensive Study On The Applications Of Fractional Chromatic Numbers

Abstract

In “the investigation of fractional chromatic numbers within the domain of graph theory, seminal references provide extensive insights. Zhang's comprehensive survey published in 2009 presents an elaborate synthesis of definitions, methodologies, and applications pertinent to this area of study. The foundational textbook authored by Chartrand and Lesniak (2005) functions as a pivotal resource for grasping fundamental concepts inherent to graph theory, while West's introductory volume (2001) imparts essential knowledge regarding graph structures and their properties. The monograph by Chen and Zhu (2009) thoroughly examines fractional graph theory, proffering a logical framework for understanding graph theory principles, inclusive of fractional chromatic numbers. Diestel's extensive textbook (2005) encompasses a multitude of facets pertaining to graph theory, notably including graph coloring and fractional chromatic numbers. The research conducted by Guan and Zhu (2011) investigates fractional chromatic numbers within the framework of Mycielski's theorem and its subsequent generalizations. Furthermore, the contributions of Haas, Nowakowski, and Rousseau (2002) yield insights into the fractional chromatic number associated with various graphs. Kostochka and Yancey (2003) scrutinize the fractional chromatic number of triangle-free subcubic graphs, while the study by Chen, Kierstead, and Zhu (2006) delves into this numerical characteristic concerning triangle-free graphs characterized by substantial minimum degree. Caro, Lévêque, and Reed (2007) engage with a generalization of Reed's conjecture, whereas Havet and Sereni (2008) explore the interplay between fractional chromatic numbers, odd holes, and perfect graphs. Finally, Nešetřil and Raspaud (1999) enhance the comprehension of the fractional chromatic number associated with cubic graphs.”