Broadcast Domination in Subcubic Graphs

For a graph G, a function f : V(G) ->{0, 1, 2,..., diam(G)} is called a dominating broadcast on G if for each vertex u is an element of V(G), there exists a vertex v in G with f(nu) > 0 such that d(G)(u, v) <= f(nu), where d(G)(u, nu) denotes the distance between u and nu in G. The cost of a dominating broadcast f is Sigma(nu is an element of V(G))f(nu). The broadcast domination number of G, denoted by gamma(b)(G), is the minimum cost of a dominating broadcast in G. For an integer k >= 1, a function f : V(G) -> {0, 1, 2,..., k} is called a k-limited dominating broadcast in G if for each vertex u is an element of V(G), there exists a vertex v in G with f(nu) > 0 such that d(G)(u, nu) <= f(nu). The cost of a k-limited dominating broadcast f is Sigma(nu is an element of V(G))integral(nu). The k-limited broadcast domination number of G, denoted by gamma(b,k)(G), 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 gamma(b,2)(G) <= n/3. In this paper, we show that if G is a cubic graph on n vertices, then gamma(b)(G) <= n/3, and this bound is sharp.