{"paper":{"title":"On the maximum order of graphs embedded in surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David R. Wood, Eran Nevo, Guillermo Pineda-Villavicencio","submitted_at":"2013-12-06T02:24:40Z","abstract_excerpt":"The maximum number of vertices in a graph of maximum degree $\\Delta\\ge 3$ and fixed diameter $k\\ge 2$ is upper bounded by $(1+o(1))(\\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance, for the class of trees there is an upper bound of $(2+o(1))(\\Delta-1)^{\\lfloor k/2\\rfloor}$ for a fixed $k$. The main result of this paper is that graphs embedded in surfaces of bounded Euler genus $g$ behave like trees, in the sense that, for large $\\Delta$, such graphs have orders bounded from above by \\[begin{cases} c(g+1)(\\Delta-1)^{\\lfloor k/2\\rfloor} & "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.1753","kind":"arxiv","version":2},"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"}