Optimal bounds on codes for location in circulant graphs

TL;DR

This paper investigates optimal bounds for locating, identifying, and self-identifying codes in specific circulant graphs, introducing new methods and providing exact values for certain cases.

Contribution

It develops a new grid-based method to establish lower bounds and determines exact optimal codes for particular circulant graphs, extending previous research.

Findings

Lower bounds for codes are established using grid methods.

Bounds are tight for infinitely many parameters n and d.

Exact optimal codes are found for C_n(1,3) and C_n(1,4).

Abstract

Identifying and locating-dominating codes have been studied widely in circulant graphs of type $C_n(1,2,3,\dots, r)$ over the recent years. In 2013, Ghebleh and Niepel studied locating-dominating and identifying codes in the circulant graphs $C_n(1,d)$ for $d=3$ and proposed as an open question the case of $d > 3$. In this paper we study identifying, locating-dominating and self-identifying codes in the graphs $C_n(1,d)$, $C_n(1,d-1,d)$ and $C_n(1,d-1,d,d+1)$. We give a new method to study lower bounds for these three codes in the circulant graphs using suitable grids. Moreover, we show that these bounds are attained for infinitely many parameters $n$ and $d$. In addition, new approaches are provided which give the exact values for the optimal self-identifying codes in $C_n(1,3)$ and $C_n(1,4).$