Layer-Shift Equitable Coloring of Pan Graphs and Their Cartesian Products with Paths: Theory and Visualization

We determine the equitable chromatic number of the pan graph C (+1) n and its Cartesian product with a path Pm.We prove that χ= C (+1) n = 2 when n is even and χ= C (+1) n = 3 when n is odd, and show that the same values hold for the Cartesian product C (+1) n Pm for all m ≥ 2. The proof uses a layer-shift construction that extends an equitable coloring along the path without increasing the number of colors.A second contribution of this work is an interactive visualization framework implemented in Python.The tool generates pan and pan-path graphs, applies the equitable coloring algorithm step by step, and produces both static snapshots and animations.This visualization supports verification of our results and enhances conceptual understanding of independence, equitability, and graph-product structure.The framework strengthens mathematics education through accessible computational exploration and directly supports Sustainable Development Goal (SDG) 4: Quality Education, particularly in advancing technology-enhanced learning in higher mathematics.