On the broadcast independence number of circulant graphs

Abstract

An independent broadcast on a graph [Formula: see text] is a function [Formula: see text] such that (i) [Formula: see text] for every vertex [Formula: see text], where [Formula: see text] denotes the diameter of [Formula: see text] and [Formula: see text] the eccentricity of vertex [Formula: see text], and (ii) [Formula: see text] for every two distinct vertices [Formula: see text] and [Formula: see text] with [Formula: see text]. The broadcast independence number [Formula: see text] of [Formula: see text] is then the maximum value of [Formula: see text], taken over all independent broadcasts on [Formula: see text]. We prove that every circulant graph of the form [Formula: see text], [Formula: see text], admits an optimal [Formula: see text]-bounded independent broadcast, that is, an independent broadcast [Formula: see text] satisfying [Formula: see text] for every vertex [Formula: see text], except when [Formula: see text], or [Formula: see text] and [Formula: see text] is even. We then determine the broadcast independence number of various classes of such circulant graphs, and prove in particular that [Formula: see text], except for [Formula: see text], [Formula: see text], or [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the independence number of [Formula: see text].