feat(Arxiv): update 2107.00295 Conjecture 1.6 status - #5105
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@guelmi94 could you review this? (Adding you because you originally added this conjecture..) |
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Conjecture 1.6 of Cho–Choi–Park (arxiv/2107.00295), formalized in this file as
independentDominationEvenandindependentDominationOdd, has been proved in the literature.Eun-Kyung Cho, Jinha Kim, Minki Kim and Sang-il Oum, Independent domination of graphs with bounded maximum degree, J. Combin. Theory Ser. B 158 (2023), 341–352 (arxiv/2202.09594, doi:10.1016/j.jctb.2022.10.004) prove, as Corollary 1.3: every graph$G$ with maximum degree at most $\Delta$ and no isolated vertices has an independent dominating set of size at most $\left(1 - \frac{\Delta}{\lfloor (\Delta+2)^2/4 \rfloor}\right)\lvert V(G)\rvert$ . The paper states that this corollary "was conjectured by Cho, Choi, and Park".
Expanding the floor by parity: for even$\Delta = D$ , $\lfloor (D+2)^2/4 \rfloor = (D+2)^2/4$ , and the bound becomes $(D+2)^2 \cdot i(G) \le (D^2+4) \cdot n$ ; for odd $\Delta = D$ , $\lfloor (D+2)^2/4 \rfloor = (D+1)(D+3)/4$ , and it becomes $(D+1)(D+3) \cdot i(G) \le (D^2+3) \cdot n$ — exactly the two statements in this file.
This PR flips both tags to
research solvedand records the reference in the module and declaration docstrings, following the pattern of #4628. Proof bodies are left assorry(no formalization of the proof is claimed).