{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T01:53:45Z","timestamp":1777427625825,"version":"3.51.4"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A $d$-defective $k$-painting game on a graph $G$ is played by two players: Lister and Painter.\u00a0Initially, each vertex is uncolored and has $k$ tokens.\u00a0In each round, Lister marks a chosen set $M$ of uncolored vertices and removes one token from each marked vertex.\u00a0In response, Painter colors vertices in a subset $X$ of $M$ which induce a subgraph $G[X]$ of maximum degree at most $d$.\u00a0Lister wins the game if at the end of some round there is an uncolored vertex that has no more tokens left.\u00a0Otherwise, all vertices eventually get colored and Painter wins the game.\u00a0We say that $G$ is $d$-defective $k$-paintable if Painter has a winning strategy in this game.\u00a0In this paper we show that every planar graph is 3-defective 3-paintable and give a construction of a planar graph that is not 2-defective 3-paintable.<\/jats:p>","DOI":"10.37236\/7084","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:33:17Z","timestamp":1578670397000},"source":"Crossref","is-referenced-by-count":3,"title":["Defective 3-Paintability of Planar Graphs"],"prefix":"10.37236","volume":"25","author":[{"given":"Grzegorz","family":"Gutowski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ming","family":"Han","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tomasz","family":"Krawczyk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xuding","family":"Zhu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2018,5,25]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i2p34\/pdf_1","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i2p34\/pdf_1","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:33:37Z","timestamp":1579235617000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i2p34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,25]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,4,13]]}},"URL":"https:\/\/doi.org\/10.37236\/7084","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,5,25]]},"article-number":"P2.34"}}