{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:59Z","timestamp":1753893839946,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The product dimension of a graph $G$ is defined as the minimum natural number $l$ such that $G$ is an induced subgraph of a direct product of $l$ complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and $k$-degenerate graphs. We show that every forest on $n$ vertices has product dimension at most $1.441 \\log n + 3$. This improves the best known upper bound of $3 \\log n$ for the same due to Poljak and Pultr. The technique used in arriving at the above bound is extended and combined with a well-known result on the existence of orthogonal Latin squares to show that every graph on $n$ vertices with treewidth at most $t$ has product dimension at most $(t+2)(\\log n + 1)$. We also show that every $k$-degenerate graph on $n$ vertices has product dimension at most $\\lceil 5.545 k \\log n \\rceil + 1$. This improves the upper bound of\u00a0 $32 k \\log n$ for the same by Eaton and\u00a0 R\u0151dl.<\/jats:p>","DOI":"10.37236\/2698","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:22:13Z","timestamp":1578705733000},"source":"Crossref","is-referenced-by-count":0,"title":["Product Dimension of Forests and Bounded Treewidth Graphs"],"prefix":"10.37236","volume":"20","author":[{"given":"L. Sunil","family":"Chandran","sequence":"first","affiliation":[]},{"given":"Rogers","family":"Mathew","sequence":"additional","affiliation":[]},{"given":"Deepak","family":"Rajendraprasad","sequence":"additional","affiliation":[]},{"given":"Roohani","family":"Sharma","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,9,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i3p42\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i3p42\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:13:31Z","timestamp":1579259611000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i3p42"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,9,20]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2013,7,19]]}},"URL":"https:\/\/doi.org\/10.37236\/2698","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,9,20]]},"article-number":"P42"}}