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Abstract
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A total Roman dominating function on a graph G is a labeling f : V (G) → {0, 1, 2} such
that every vertex with label 0 has a neighbor with label 2 and the subgraph of G induced
by the set of all vertices of positive weight has no isolated vertex. A set {f1, f2, . . . , fd}
of total Roman dominating functions on G with the property that Pd
i=1 fi(v) ≤ 2 for
each v ∈ V (G), is called a total Roman dominating family (of functions) on G. The
maximum number of functions in a total Roman dominating family on G is the total
Roman domatic number of G, denoted by dtR(G). In this paper, we investigate the
properties of total Roman domatic number in graphs. In particular, we present some
sharp bounds for dtR(G) and we determine the total Roman domatic number of some
special graphs.
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