For a graph G, a function f:V(G)→{0,1,2,…,diam(G)}\documentclass[12pt]{minimal}
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\begin{document}$$f:V(G)\rightarrow \{0,1,2,\ldots , diam(G)\}$$\end{document} is called a dominating broadcast on G if for each vertex u∈V(G)\documentclass[12pt]{minimal}
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\begin{document}$$u\in V(G)$$\end{document}, there exists a vertex v in G with f(v)>0\documentclass[12pt]{minimal}
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\begin{document}$$f(v)>0$$\end{document} such that dG(u,v)≤f(v)\documentclass[12pt]{minimal}
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\begin{document}$$d_{G}(u,v)\le f(v)$$\end{document}, where dG(u,v)\documentclass[12pt]{minimal}
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\begin{document}$$d_{G}(u,v)$$\end{document} denotes the distance between u and v in G. The cost of a dominating broadcast f is ∑v∈V(G)f(v)\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{v\in V(G)}f(v)$$\end{document}. The broadcast domination number of G, denoted by γb(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _b(G)$$\end{document}, is the minimum cost of a dominating broadcast in G. For an integer k≥1\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 1$$\end{document}, a function f:V(G)→{0,1,2,…,k}\documentclass[12pt]{minimal}
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\begin{document}$$f:V(G) \rightarrow \{0,1,2,\ldots ,k\}$$\end{document} is called a k-limited dominating broadcast in G if for each vertex u∈V(G)\documentclass[12pt]{minimal}
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\begin{document}$$u\in V(G)$$\end{document}, there exists a vertex v in G with f(v)>0\documentclass[12pt]{minimal}
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\begin{document}$$f(v)>0$$\end{document} such that dG(u,v)≤f(v)\documentclass[12pt]{minimal}
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\begin{document}$$d_{G}(u,v)\le f(v)$$\end{document}. The cost of a k-limited dominating broadcast f is ∑v∈V(G)f(v)\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{v\in V(G)}f(v)$$\end{document}. The k-limited broadcast domination number of G, denoted by γb,k(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{b,k}(G)$$\end{document}, is the minimum cost of a k-limited dominating broadcast in G. Henning, MacGillivray, Yang (Discrete Appl. Math. 285 (2020) 691-706) posed a conjecture which says that G is a cubic graph on n vertices, then γb,2(G)≤n3\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{b,2}(G)\le \frac{n}{3}$$\end{document}. In this paper, we show that if G is a cubic graph on n vertices, then γb(G)≤n3\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _b(G) \le \frac{n}{3}$$\end{document}, and this bound is sharp.
