Abstract
A total Roman dominating function on a graph [Formula: see text] is a labeling [Formula: see text] such that every vertex with label 0 has a neighbor with label 2 and the subgraph of [Formula: see text] induced by the set of all vertices of positive weight has no isolated vertex. A set [Formula: see text] of total Roman dominating functions on [Formula: see text] with the property that [Formula: see text] for each [Formula: see text], is called a total Roman dominating family (of functions) on [Formula: see text]. The maximum number of functions in a total Roman dominating family on [Formula: see text] is the total Roman domatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we investigate the properties of total Roman domatic number in graphs. In particular, we present some sharp bounds for [Formula: see text] and we determine the total Roman domatic number of some special graphs.
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