{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:09Z","timestamp":1753893849166,"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>Let $a,b\\in\\mathbb{N}$. A graph $G$ is $(a,b)$-choosable if for any list assignment $L$ such that $|L(v)|\\ge a$, there exists a coloring in which each vertex $v$ receives a set $C(v)$ of $b$ colors such that $C(v)\\subseteq L(v)$ and $C(u)\\cap C(w)=\\emptyset$ for any $uw\\in E(G)$. In the online version of this problem, on each round, a set of vertices allowed to receive a particular color is marked, and the coloring algorithm chooses an independent subset of these vertices to receive that color. We say $G$ is $(a,b)$-paintable if when each vertex $v$ is allowed to be marked $a$ times, there is an algorithm to produce a coloring in which each vertex $v$ receives $b$ colors such that adjacent vertices receive disjoint sets of colors.We show that every odd cycle $C_{2k+1}$ is $(a,b)$-paintable exactly when it is $(a,b)$-chosable, which is when $a\\ge2b+\\lceil b\/k\\rceil$. In 2009, Zhu conjectured that if $G$ is $(a,1)$-paintable, then $G$ is $(am,m)$-paintable for any $m\\in\\mathbb{N}$. The following results make partial progress towards this conjecture. Strengthening results of Tuza and Voigt, and of Schauz, we prove for any $m \\in \\mathbb{N}$ that $G$ is $(5m,m)$-paintable when $G$ is planar. Strengthening work of Tuza and Voigt, and of Hladky, Kral, and Schauz, we prove that for any connected graph $G$ other than an odd cycle or complete graph and any $m\\in\\mathbb{N}$, $G$ is $(\\Delta(G)m,m)$-paintable.<\/jats:p>","DOI":"10.37236\/7163","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:43:24Z","timestamp":1578671004000},"source":"Crossref","is-referenced-by-count":0,"title":["Strengthening $(a,b)$-Choosability Results to $(a,b)$-Paintability"],"prefix":"10.37236","volume":"24","author":[{"given":"Thomas","family":"Mahoney","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,10,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i4p26\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i4p26\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:42:35Z","timestamp":1579236155000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i4p26"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,20]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,10,5]]}},"URL":"https:\/\/doi.org\/10.37236\/7163","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,10,20]]},"article-number":"P4.26"}}