{"paper":{"title":"Lower Bounds for the Exponential Domination Number of $C_m \\times C_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chassidy Bozeman, Derek Young, Joshua Carlson, Michael Dairyko, Michael Young","submitted_at":"2018-03-05T21:24:16Z","abstract_excerpt":"A vertex $v$ in a porous exponential dominating set assigns weight $\\left(\\tfrac{1}{2}\\right)^{dist(v,u)}$ to vertex $u$. A porous exponential dominating set of a graph $G$ is a subset of $V(G)$ such that every vertex in $V(G)$ has been assigned a sum weight of at least 1. In this paper the porous exponential dominating number, denoted by $\\gamma_e^*(G)$, for the graph $G = C_m \\times C_n$ is discussed. Anderson et. al. proved that $\\frac{mn}{15.875}\\le \\gamma_e^*(C_m \\times C_n) \\le \\frac{mn}{13}$ and conjectured that $\\frac{mn}{13}$ is also the asymptotic lower bound. We use a linear program"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.01933","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}