{"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":1753893839140,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We are interested in $(\\varepsilon)$-regular bipartite graphs which are the central objects in the regularity lemma of Szemer\u00e9di for sparse graphs. A bipartite graph $G=(A\\uplus B,E)$ with density $p={|E|}\/({|A||B|})$ is $(\\varepsilon)$-regular if for all sets $A'\\subseteq A$ and $B'\\subseteq B$ of size $|A'|\\geq \\varepsilon|A|$ and $|B'|\\geq \\varepsilon |B|$, it holds that $\\left| {e_G(A',B')}\/{(|A'||B'|)}- p\\right| \\leq \\varepsilon p$. In this paper we prove a characterization for $(\\varepsilon)$-regularity. That is, we give a set of properties that hold for each $(\\varepsilon)$-regular graph, and conversely if the properties of this set hold for a bipartite graph, then the graph is $f(\\varepsilon)$-regular for some appropriate function $f$ with $f(\\varepsilon)\\rightarrow 0$ as $\\varepsilon\\rightarrow 0$. The properties of this set concern degrees of vertices and common degrees of vertices with sets of size $\\Theta(1\/p)$ where $p$ is the density of the graph in question.<\/jats:p>","DOI":"10.37236\/923","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:54:08Z","timestamp":1578700448000},"source":"Crossref","is-referenced-by-count":3,"title":["A Characterization for Sparse $\\varepsilon$-Regular Pairs"],"prefix":"10.37236","volume":"14","author":[{"given":"Stefanie","family":"Gerke","sequence":"first","affiliation":[]},{"given":"Angelika","family":"Steger","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2007,1,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1r4\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1r4\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:05:24Z","timestamp":1579302324000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v14i1r4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1,3]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2007,1,3]]}},"URL":"https:\/\/doi.org\/10.37236\/923","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2007,1,3]]},"article-number":"R4"}}