{"paper":{"title":"On Grundy total domination number in product graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bal\\'azs Patk\\'os, Bo\\v{s}tjan Bre\\v{s}ar, Csilla Bujt\\'as, Ga\\v{s}per Ko\\v{s}mrlj, M\\'at\\'e Vizer, Sandi Klav\\v{z}ar, Tanja Gologranc, Tilen Marc, Zsolt Tuza","submitted_at":"2017-12-23T14:42:14Z","abstract_excerpt":"A longest sequence $(v_1,\\ldots,v_k)$ of vertices of a graph $G$ is a Grundy total dominating sequence of $G$ if for all $i$, $N(v_i) \\setminus \\bigcup_{j=1}^{i-1}N(v_j)\\not=\\emptyset$. The length $k$ of the sequence is called the Grundy total domination number of $G$ and denoted $\\gamma_{gr}^{t}(G)$. In this paper, the Grundy total domination number is studied on four standard graph products. For the direct product we show that $\\gamma_{gr}^t(G\\times H) \\geq \\gamma_{gr}^t(G)\\gamma_{gr}^t(H)$, conjecture that the equality always holds, and prove the conjecture in several special cases. For the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08780","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"}