Damghan University PressGlobal Analysis and Discrete Mathematics2476-53417120221101Dominated Coloring of Certain Graphs899928710.22128/gadm.2022.624.1081ENFatemehChoopaniDepartment of Mathematics, Ferdowsi University of Mashhad, Mashhad, IranAbbasJafarzadehDepartment of
Mathematics, Quchan University of Technology, Quchan, IranDoost AliMojdehDepartment of Mathematics, University of Mazandaran, Babolsar, IranJournal Article20221014A proper coloring of a graph <em>G</em> is called a dominated coloring whenever each color class is dominated by at least one vertex. The minimum number of colors among all dominated colorings of <em>G</em> is called its dominated chromatic number, denoted by <em>χ<sub>dom</sub>(G)</em>. We define a parameter related to dominated coloring, namely dominated chromatic covering. For a minimum dominated coloring of <em>G</em>, a set of vertices <em>S</em> is called a dominated chromatic covering if each color class is dominated by a vertex of <em>S</em>. The minimum cardinality of a dominated chromatic covering of <em>G</em> is called its dominated chromatic covering number, denoted by <em>θ<sub>χdom</sub>(G)</em> . It is clear that <em>θ<sub>χdom</sub>(G)</em> ≤ <em>χ<sub>dom</sub>(G)</em>. In this paper, we obtain the dominated chromatic number and <em>θ<sub>χdom</sub>(G)</em> when <em>G</em> is middle and total graph of paths and cycles.https://gadm.du.ac.ir/article_287_6ee28d4d597d24bb863fc400911de72a.pdf