On the Broadcast Independence Number of Circulant Graphs
Résumé
An independent broadcast on a graph G is a function f : V −→ {0,. .. , diam(G)} such that (i) f (v) ≤ e(v) for every vertex v ∈ V (G), where diam(G) denotes the diameter of G and e(v) the eccentricity of vertex v, and (ii) d(u, v) > max{f (u), f (v)} for every two distinct vertices u and v with f (u)f (v) > 0. The broadcast independence number β b (G) of G is then the maximum value of v∈V f (v), taken over all independent broadcasts on G. We prove that every circulant graph of the form C(n; 1, a), 3 ≤ a ≤ ⌊ n 2 ⌋, admits an optimal 2-bounded independent broadcast, that is, an independent broadcast f satisfying f (v) ≤ 2 for every vertex v, except when n = 2a + 1, or n = 2a and a is even. We then determine the broadcast independence number of various classes of such circulant graphs, and prove that, for most of these classes, the equality β b (C(n; 1, a)) = α(C(n; 1, a)) holds, where α(C(n; 1, a)) denotes the independence number of C(n; 1, a).
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