Empirical Convergence of the Chromatic-Number–Connectivity Gap in Random Geometric Graphs

An exact computational study of random geometric graphs comparing finite-n chromatic-number and connectivity thresholds against known asymptotic theory (Penrose 1999; McDiarmid & Müller 2011; Ravelomanana 2017). Runs exact backtracking graph coloring across n=10–40 (100 trials per n) to measure how often the chromatic number reaches a target value before the graph connects, and how the empirical connectivity threshold converges toward the theoretical asymptotic formula r_c(n) = √(ln n / (πn)). Includes full source code, raw results, and a written report with methodology, results, discussion, and limitations.