{"paper":{"title":"On the total neighbour sum distinguishing index of graphs with bounded maximum average degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Herv\\'e Hocquard, Jakub Przyby{\\l}o","submitted_at":"2018-03-05T20:11:39Z","abstract_excerpt":"A proper total $k$-colouring of a graph $G=(V,E)$ is an assignment $c : V \\cup E\\to \\{1,2,\\ldots,k\\}$ of colours to the edges and the vertices of $G$ such that no two adjacent edges or vertices and no edge and its end-vertices are associated with the same colour. A total neighbour sum distinguishing $k$-colouring, or tnsd $k$-colouring for short, is a proper total $k$-colouring such that $\\sum_{e\\ni u}c(e)+c(u)\\neq \\sum_{e\\ni v}c(e)+c(v)$ for every edge $uv$ of $G$. We denote by $\\chi''_{\\Sigma}(G)$ the total neighbour sum distinguishing index of $G$, which is the least integer $k$ such that a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.02686","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"}