{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T06:14:44Z","timestamp":1762409684838,"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>A graph $G$ is $(d_1,...,d_l)$-colorable if the vertex set of\u00a0$G$ can be partitioned into subsets $V_1,\\ldots ,V_l$ such that the\u00a0graph $G[V_i]$ induced by the vertices of $V_i$ has maximum degree at\u00a0most $d_i$ for all $1 \\leq i \\leq l$. In this paper, we focus on\u00a0complexity aspects of such colorings when $l=2,3$. More precisely, we\u00a0prove that, for any fixed integers $k,j,g$ with $(k,j) \\neq (0,0)$ and\u00a0$g\\geq3$, either every planar graph with girth at least $g$ is\u00a0$(k,j)$-colorable or it is NP-complete to determine whether a planar\u00a0graph with girth at least $g$ is $(k,j)$-colorable. Also, for any\u00a0fixed integer $k$, it is NP-complete to determine whether a planar\u00a0graph that is either $(0,0,0)$-colorable or non-$(k,k,1)$-colorable is\u00a0$(0,0,0)$-colorable. Additionally, we exhibit non-$(3,1)$-colorable\u00a0planar graphs with girth 5 and non-$(2,0)$-colorable planar graphs\u00a0with girth 7.\u00a0<\/jats:p>","DOI":"10.37236\/3509","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T19:30:43Z","timestamp":1578684643000},"source":"Crossref","is-referenced-by-count":20,"title":["Near-Colorings: Non-Colorable Graphs and NP-Completeness"],"prefix":"10.37236","volume":"22","author":[{"given":"M.","family":"Montassier","sequence":"first","affiliation":[]},{"given":"P.","family":"Ochem","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,3,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p57\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p57\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:24:56Z","timestamp":1579238696000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i1p57"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3,6]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/3509","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,3,6]]},"article-number":"P1.57"}}