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Regular graphs with large Italian domatic number | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 22، دوره 7، شماره 2، اسفند 2022، صفحه 257-271 اصل مقاله (508.12 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2021.27092.1194 | ||
| نویسنده | ||
| Jeremy Lyle* | ||
| Department of Mathematics and Computer Science, Olivet Nazarene University, Bourbonnais, IL | ||
| چکیده | ||
| 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. 258 (2019), 235--241], Volkmann defined the 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. | ||
| کلیدواژهها | ||
| Italian domination؛ Nordhaus-Gaddum؛ Domatic number | ||
| مراجع | ||
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[12] L. Volkmann, The Roman {2}-domatic number of graphs, Discrete Appl. Math. 258 (2019), 235–241. | ||
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