{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:09:02Z","timestamp":1753880942301,"version":"3.41.2"},"reference-count":9,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2023,5]]},"abstract":"<jats:p> Let [Formula: see text] be a graph. A [Formula: see text]-coloring of [Formula: see text] is a mapping [Formula: see text], if each color class induces a [Formula: see text]-free subgraph. For a graph [Formula: see text] of order at least [Formula: see text], a [Formula: see text]-free [Formula: see text]-coloring of [Formula: see text], is a mapping [Formula: see text], so that the induced subgraph by each color class of [Formula: see text], contains no copy of [Formula: see text]. The [Formula: see text]-free chromatic number of [Formula: see text], is the minimum number [Formula: see text], so that it has a [Formula: see text]-free [Formula: see text]-coloring, and denoted by [Formula: see text]. Suppose that [Formula: see text] be a family of graphs, we say a graph [Formula: see text] has a [Formula: see text]-free [Formula: see text]-coloring, if there exists a map [Formula: see text], such that each color class [Formula: see text] does not contain any members of [Formula: see text]. In this paper, we give some bounds and attributes on the [Formula: see text]-free chromatic number of graphs in terms of the number of vertices, maximum degree, minimum degree, and chromatic number. Our main results are the Nordhaus\u2013Gaddum-type theorem for the [Formula: see text]-free chromatic number of a graph. <\/jats:p>","DOI":"10.1142\/s1793830922501142","type":"journal-article","created":{"date-parts":[[2022,6,4]],"date-time":"2022-06-04T15:30:31Z","timestamp":1654356631000},"source":"Crossref","is-referenced-by-count":0,"title":["Nordhaus\u2013Gaddum problem in terms of G-free coloring"],"prefix":"10.1142","volume":"15","author":[{"given":"Yaser","family":"Rowshan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran"}]}],"member":"219","published-online":{"date-parts":[[2022,7,8]]},"reference":[{"issue":"4","key":"S1793830922501142BIB001","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1016\/j.dam.2011.12.018","volume":"161","author":"Aouchiche M.","year":"2013","journal-title":"Discr. Appl. 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