{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:46:38Z","timestamp":1759063598641,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A set of vertices in a graph is connected if it induces a connected subgraph. Using Shearer's entropy lemma and a computer search, we show that the number of connected sets in a graph with $n$ vertices and maximum vertex degree $d$ is at most $O(1.9351^n)$ for $d=3$, $O(1.9812^n)$ for $d=4$, and $O(1.9940^n)$ for $d=5$. Dually, we construct infinite families of graphs where the number of connected sets is at least $\\Omega(1.7651^n)$ for $d=3$, $\\Omega(1.8925^n)$ for $d=4$, and $\\Omega(1.9375^n)$ for $d=5$.<\/jats:p>","DOI":"10.37236\/7462","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:01:26Z","timestamp":1578668486000},"source":"Crossref","is-referenced-by-count":3,"title":["On the Number of Connected Sets in Bounded Degree Graphs"],"prefix":"10.37236","volume":"25","author":[{"ORCID":"https:\/\/2.zoppoz.workers.dev:443\/https\/orcid.org\/0000-0002-5262-7919","authenticated-orcid":false,"given":"Kustaa","family":"Kangas","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Petteri","family":"Kaski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Janne H.","family":"Korhonen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mikko","family":"Koivisto","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2018,11,16]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i4p34\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i4p34\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:21:31Z","timestamp":1579234891000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i4p34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,16]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,10,5]]}},"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.37236\/7462","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2018,11,16]]},"article-number":"P4.34"}}