Six Colors for Generalized Sierpiński Graphs Over C5 and a Sharp Classification of Path Products
Year: 2026
Type: preprint
Abstract: We prove that the packing chromatic number of the generalized Sierpiński graph over the five-cycle is six in every dimension at least two. Together with the known values for the other cycle bases of length at least four, this completes their classification. Two explicit 125-vertex templates and twenty oriented profile states provide the construction. Each state admits two independently selectable productions, yielding at least 20 · 2^((5^(n−3)−1)/4) optimal labelled colorings in dimension n ≥ 3; their normalized base-two logarithmic count has a limit of at least 1/500. For Sierpiński path products, we determine the maximum packing chromatic number throughout the previously unresolved range m ≥ 12, n ≥ 3: it is seven exactly when m ≥ 20, n ≥ 5, and six otherwise. The proof gives two sharp structural guarantees for trees with at most two pendant paths at each spine vertex: nineteen spine vertices, or sixteen vertices bearing two arms of length at least two, suffice for six colors. Explicit examples show both guarantees are sharp. The finite computations have standalone certificates; the unbounded claims follow from the stated constructions, closure arguments, and induction.
Source: Preprints.org
Author: Xinyao Xu
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225558165