Regular graphs with large Italian domatic number

Abstract

For a graph $G$, an Italian dominating function is a function $f: V(G) rightarrow {0,1,2}$ such that for each vertex $v in V(G)$ either $f(v) neq 0$, or $sum_{u in N(v)} f(u) geq 2$.If a family $mathcal{F} = {f_1, f_2, dots, f_t}$ of distinct Italian dominating functions satisfy $sum^t_{i = 1} f_i(v) leq 2$ for each vertex $v$, then this is called an Italian dominating family.In [L. Volkmann, The {R}oman {${2}$}-domatic number of graphs, Discrete Appl. Math. {bf 258} (2019), 235--241], Volkmann defined the textit{Italian domatic number} of $G$, $d_{I}(G)$, as the maximum cardinality of any Italian dominating family. In this same paper, questions were raised about the Italian domatic number of regular graphs. In this paper, we show that two of the conjectures are false, and examine some exceptions to a Nordhaus-Gaddum type inequality.