Broadcast independence number of oriented circulant graphs

TL;DR

This work addresses the broadcast independence number for oriented circulant graphs $\overrightarrow{C}(n;1,a)$. By formalizing independent broadcasts with $f:V\to\{0,\dots,\operatorname{diam}(\overrightarrow{G})\}$ and using the relation $\beta_b(\overrightarrow{G})\ge \max\{\beta_b(G),\operatorname{diam}(\overrightarrow{G})\}$, the authors develop an orientation-aware framework and exploit isomorphisms to reduce the parameter space. They establish exact values for several base cases and then develop a general methodology that identifies $\ell$-bounded and $q$-bounded optimal broadcasts, deriving diameter bounds and partition-based cost bounds via $A_f^i$ and $B_f^j$. Building on this framework, they obtain broad upper and lower bounds for $\beta_b$ in families determined by divisibility relations between $n$ and $a$, and present numerous exact values for families such as $n=k(a-1)$, $n=qa$, and $n=qa+a-1$, including isomorphism-induced equalities, thereby advancing the understanding of broadcast independence on oriented circulants.

Abstract

In 2001, D. Erwin \cite{Erw01} introduced in his Ph.D. dissertation the notion of broadcast independence in unoriented graphs. Since then, some results but not many, are published on this notion, including research work on the broadcast independence number of unoriented circulant graphs \cite{LBS23}. In this paper, we are focused in the same parameter but of the class of oriented circulant graphs. An independent broadcast on an oriented graph $\overrightarrow{G}$ is a function $f: V\longrightarrow \{0,\ldots,\diam(\overrightarrow{G})\}$ such that $(i)$ $f(v)\leq e(v)$ for every vertex $v\in V(\overrightarrow{G})$, where $\diam(\overrightarrow{G})$ denotes the diameter of $\overrightarrow{G}$ and $e(v)$ the eccentricity of vertex $v$, and $(ii)$ $d_{\overrightarrow{G}}(u,v) > f(u)$ for every distinct vertices $u$, $v$ with $f(u)$, $f(v)>0$, where $d_{\overrightarrow{G}}(u,v)$ denotes the length of a shortest oriented path from $u$ to $v$. The broadcast independence number $β_b(\overrightarrow{G})$ of $\overrightarrow{G}$ is then the maximum value of $\sum_{v \in V} f(v)$, taken over all independent broadcasts on $\overrightarrow{G}$. The goal of this paper is to study the properties of independent broadcasts of oriented circulant graphs $\overrightarrow{C}(n;1,a)$, for any integers $n$ and $a$ with $n>|a|\geq 1$ and $a \notin \{1,n-1\}$. Then, we give some bounds and some exact values for the number $β_b(\overrightarrow{C}(n;1,a))$.

Figures (8)

• Figure 1: Two independent broadcasts $f_1$ on $\overrightarrow{C}(12;1,4)$ and $f_2$ on $\overrightarrow{C}(12;1,2)$.
• Figure 2: The set $B_f^j$ (the black vertices) with $n=16$ and $a=8$.
• Figure 3: The set $L(v_i)$ (black vertex and grey vertices), with $a=7$ and $f(v_i)=4$.
• Figure 5: The isomorphic graphs $\overrightarrow{C}(9;1,2)$ and $\overrightarrow{C}(9;1,5)$.
• Figure 8: The circulant graph $\overrightarrow{C}(qa;1,a)$ with $q=3$ and $a=7$

Theorems & Definitions (29)

• Proposition 1
• Theorem 3
• Theorem 4
• Theorem 5
• Corollary 6
• Theorem 7
• Proposition 8
• Proposition 9
• Proposition 10
• Lemma 11