Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 7 1 2022 11 01 Dominated Coloring of Certain Graphs 89 99 287 10.22128/gadm.2022.624.1081 EN Fatemeh Choopani Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran Abbas Jafarzadeh Department of Mathematics, Quchan University of Technology, Quchan, Iran Doost Ali Mojdeh Department of Mathematics, University of Mazandaran, Babolsar, Iran Journal Article 2022 10 14 A proper coloring of a graph G 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 G is called its dominated chromatic number, denoted by χdom(G). We define a parameter related to dominated coloring, namely dominated chromatic covering. For a minimum dominated coloring of G, a set of vertices S is called a dominated chromatic covering if each color class is dominated by a vertex of S. The minimum cardinality of a dominated chromatic covering of G is called its dominated chromatic covering number, denoted by θχdom(G) . It is clear that θχdom(G) ≤ χdom(G). In this paper, we obtain the dominated chromatic number and θχdom(G) when G is middle and total graph of paths and cycles.

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