Commuting Conjugacy Class Graphs of Symmetric Groups

Abstract

For a finite group, the commuting conjugacy class graph has the non-central conjugacy classes as its vertices, with two classes adjacent when they contain commuting representatives. This essay studies the graph for symmetric groups, determining exactly when it is connected, bounding the diameter of its principal component, and obtaining a linear lower bound for its clique and chromatic numbers. It also classifies when the graph is perfect and records further computational evidence and conjectures.