{"paper":{"title":"On the semi-proper orientations of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Ali Dehghan","submitted_at":"2019-05-08T02:05:22Z","abstract_excerpt":"A {\\it semi-proper orientation} of a given graph $G$ is a function $(D,w)$ that assigns an orientation $D(e)$ and a positive integer weight $ w(e)$ to each edge $e$ such that for every two adjacent vertices $v$ and $u$, $S_{(D,w)}(v) \\neq S_{(D,w)}(u) $, where $S_{(D,w)}(v) $ is the sum of the weights of edges with head $v$ in $D$. The {\\it semi-proper orientation number} of a graph $G$, denoted by $\\overrightarrow{\\chi}_s (G)$, is $ \\min_{(D,w)\\in \\Gamma} \\max_{v\\in V(G)} S_{(D,w)}(v) $, where $\\Gamma$ is the set of all semi-proper orientations of $G$. The {\\it optimal semi-proper orientation"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.02867","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"}