{"paper":{"title":"On the number of cycles in a graph with restricted cycle lengths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bal\\'azs Keszegh, Bal\\'azs Patk\\'os, Cory Palmer, D\\'aniel Gerbner","submitted_at":"2016-10-11T19:48:27Z","abstract_excerpt":"Let $L$ be a set of positive integers. We call a (directed) graph $G$ an $L$\\emph{-cycle graph} if all cycle lengths in $G$ belong to $L$. Let $c(L,n)$ be the maximum number of cycles possible in an $n$-vertex $L$-cycle graph (we use $\\vec{c}(L,n)$ for the number of cycles in directed graphs). In the undirected case we show that for any fixed set $L$, we have $c(L,n)=\\Theta_L(n^{\\lfloor k/\\ell \\rfloor})$ where $k$ is the largest element of $L$ and $2\\ell$ is the smallest even element of $L$ (if $L$ contains only odd elements, then $c(L,n)=\\Theta_L(n)$ holds.) We also give a characterization of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.03476","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"}