{"paper":{"title":"On the maximum number of edges in plane graph with fixed exterior face degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adem Kilicman, Gek L. Chiab, Hazim Michman Trao, Niran Abbas Ali","submitted_at":"2017-08-07T07:53:44Z","abstract_excerpt":"A well known Euler's formula consequence's corollary in graph theory states that: For a connected simple planar graph with $n$ vertices and $m$ edges, and girth $g$, we have $m \\leq \\frac{g}{g-2}(n-2)$. We show that a connected simple plane graph with $n$ vertices and girth $g$, and exterior face of degree $h$ has at most $\\frac{g}{g-2}(n-2)- \\frac{1}{g-2}(h-g)$ edges. A \\emph{convex hull $g$-angulation} is a connected plane graph in which the exterior face is a simple $h$-cycle and all inner faces are $g$-cycles. For a given set $S$ of $n$ point in the plane having $h$ points in the boundary "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.02024","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"}