On the Broadcast Independence Number of Circulant Graphs

TL;DR

This paper investigates the broadcast independence number of circulant graphs, establishing optimal bounds and conditions for specific graph classes, and compares it with the traditional independence number.

Contribution

It proves that most circulant graphs of the form C(n;1,a) admit a 2-bounded optimal independent broadcast and determines their broadcast independence number.

Findings

Most C(n;1,a) graphs have equal broadcast independence and independence numbers.

Optimal 2-bounded independent broadcasts exist for these graphs.

Explicit values of broadcast independence numbers are derived for various classes.

Abstract

An independent broadcast on a graph $G$ is a function $f: V \longrightarrow \{0,\ldots,{\rm diam}(G)\}$ such that $(i)$ $f(v)\leq e(v)$ for every vertex $v\in V(G)$, where $\operatorname{diam}(G)$ 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 $\sum_{v \in V} f(v)$, taken over all independent broadcasts on $G$. We prove that every circulant graph of the form $C(n;1,a)$, $3\le a\le \lfloor\frac{n}{2} \rfloor$, admits an optimal $2$-bounded independent broadcast, that is, an independent broadcast~$f$ satisfying $f(v)\le 2$ 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 $\beta_b(C(n;1,a)) = \alpha(C(n;1,a))$ holds, where $\alpha(C(n;1,a))$ denotes the independence number of $C(n;1,a)$.