{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:16Z","timestamp":1753893796077,"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>This paper introduces the concept of weak $(d,h)$-decomposition of a graph $G$, which is defined as a partition of $E(G)$ into two subsets $E_1,E_2$, such that $E_1$ induces a $d$-degenerate graph $H_1$ and $E_2$ induces a subgraph $H_2$ with $\\alpha(H_1[N_{H_2}(v)]) \\le h$ for any vertex $v$. We prove that each planar graph admits a weak $(2,3)$-decomposition. As a consequence, every planar graph $G$ has a subgraph $H$ such that $G-E(H)$ is $3$-paintable and any proper coloring of $G-E(H)$ is a $3$-defective coloring of $G$. This improves the result in [G. Gutowski, M. Han, T. Krawczyk, and X. Zhu, Defective $3$-paintability of planar graphs, Electron. J. Combin., 25(2):\\#P2.34, 2018] that every planar graph is 3-defective $3$-paintable.<\/jats:p>","DOI":"10.37236\/11774","type":"journal-article","created":{"date-parts":[[2023,10,19]],"date-time":"2023-10-19T18:02:36Z","timestamp":1697738556000},"source":"Crossref","is-referenced-by-count":0,"title":["Weak (2,3)-Decomposition of Planar Graphs"],"prefix":"10.37236","volume":"30","author":[{"given":"Ming","family":"Han","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xuding","family":"Zhu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2023,10,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i4p14\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i4p14\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,19]],"date-time":"2023-10-19T18:02:36Z","timestamp":1697738556000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v30i4p14"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,20]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,10,6]]}},"URL":"https:\/\/doi.org\/10.37236\/11774","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2023,10,20]]},"article-number":"P4.14"}}