On the broadcast independence number of circulant graphs

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 is an element of 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 beta(b)(G) of G is then the maximum value of Sigma(v is an element of 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 <= left perpendicularn/2right perpendicular, 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 in particular that beta(b)(C(n; 1, a)) = alpha(C(n; 1, a)), except for C(n; 1, 2), C(2a + 1; 1, a), or C(2a; 1, a) with a not equal 2(p) and p >= 0, where alpha(C(n; 1, a)) denotes the independence number of C(n; 1, a).