A study on chromatic polynomials of bunch graphs

The chromatic polynomial P_G ( q ) counts the proper q -colorings of a graph and is a standard tool for measuring feasible resource assignments. This paper studies chromatic polynomials for heterogeneous bunch graphs B=H⊙(F₁,…,F_p), where a base graph H models the bunch layer and each vertex b_i hosts a possibly different local graph F_i . The basic corona-type factorization P_B(q)=P_H(q)∏i=1pPF_i(q−1) is used as a principle for graph-polynomial analysis. The new results derived from this principle include coefficient formulas, Tutte-specialization formulas, a multi-level factorization, and related algorithms. A small experiment evaluates the full graph from one six-vertex base computation and six eight-vertex local computations, giving P_B(6)=5,087,925,633,024,000,000,000,000,000,000,000. In a five-site cloud-edge maintenance instance, four windows give 725, 594, 112 feasible schedules, five windows give 4, 334, 215, 495, 680 schedules, and a local policy change is updated by replacing one factor, reducing the five-window schedule count by 71.43%. These computations show that bunch structure can turn a global chromatic-polynomial evaluation into independent local evaluations with direct operational meaning.