Global dominator coloring of unicyclic graphs

Abstract

Let [Formula: see text]. A vertex [Formula: see text] is a dominator of [Formula: see text] if [Formula: see text] dominates every vertex in [Formula: see text] and [Formula: see text] is said to be an anti-dominator of [Formula: see text] if [Formula: see text] dominates none of the vertices of [Formula: see text]. Let [Formula: see text] be a coloring of [Formula: see text] and let [Formula: see text]. A color class [Formula: see text] is called a dom-color class or an anti-dom-color class of the vertex [Formula: see text] according as [Formula: see text] is a dominator of [Formula: see text] or an anti-dominator of [Formula: see text]. The coloring [Formula: see text] is called a global dominator coloring of [Formula: see text] if every vertex of [Formula: see text] has a dom-color class and an anti-dom-color class in [Formula: see text]. The minimum number of colors required for a global dominator coloring of [Formula: see text] is called the global dominator chromatic number and is denoted by [Formula: see text]. This parameter was introduced in [I. S. Hamid and M. Rajeswari, Global dominator coloring of graphs, Discuss. Math. Graph Theory 39 (2019) 325–339]. This paper further extends the study by obtaining the value of [Formula: see text] for unicyclic graphs.