Weak Saturation of the Cube in the Clique: Exact Values for n ≤ 13 and Linear Bounds
Year: 2026
Type: preprint
Abstract: For graphs G and H, wsat(G, H) is the least number of edges of a spanning subgraph F of G such that the edges of G missing from F can be added one at a time, each completing a new copy of H. Morrison and Noel asked (Open Problem Garden, 2016) to determine wsat(K_n, Q_3), where Q_3 is the cube. We give a partial answer. For n ≤ 7 no edge can be added and the value is n(n − 1)/2. By exhaustive computer searches, each carried out with at least two independent programs, wsat(K_n, Q_3) = 15, 16, 18, 19, 21, 23 for n = 8, 9, 10, 11, 12, 13; extremal graphs are given explicitly, with certificates that can be checked without a computer. The values for n ≤ 11 and the upper bound wsat(K_n, Q_3) ≤ ⌊(7n + 2)/4⌋ for n ≥ 9, which comes from a gadget with four vertices and seven edges, were obtained earlier in two unrefereed preprints (Carptopus, 2026; V. Saha and N. Saha, 2026); the values for n = 12, 13 are new. We prove wsat(K_n, Q_3) ≥ (5n − 7)/3 for n ≥ 8 by the cluster method of Terekhov and Zhukovskii, with clusters that may share a vertex. Hence the limit of wsat(K_n, Q_3)/n, which exists by a result of Alon, lies in [5/3, 7/4]; the lower end improves the value 11/7 given by a general theorem of Terekhov and Zhukovskii. We conjecture, on computational evidence only, that wsat(K_n, Q_3) = ⌊(7n + 2)/4⌋ for all n ≥ 9. The exact value for n ≥ 14 remains open. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OPG-60013 (Morrison–Noel, Weak saturation of the cube in the clique, Open Problem Garden, 2016).
Author: Ferudun, Alper
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222972287
Cites:
Cited by:
Related to: