Solving Two Conjectures regarding Codes for Location in Circulant Graphs

TL;DR

This paper proves two conjectures about the minimal size of locating-dominating and identifying codes in circulant graphs C_n(1,3), using a novel approach that leverages global graph properties.

Contribution

It introduces a new method that combines global and local graph properties to determine minimal code sizes, confirming two conjectures in the field.

Findings

Confirmed the conjecture on the minimal size of locating-dominating codes in C_n(1,3).

Proved the conjecture on the minimal size of identifying codes in C_n(1,3).

Developed a novel proof technique combining global and local graph properties.

Abstract

Identifying and locating-dominating codes have been widely studied in circulant graphs of type $C_n(1,2, \ldots, r)$, which can also be viewed as power graphs of cycles. Recently, Ghebleh and Niepel (2013) considered identification and location-domination in the circulant graphs $C_n(1,3)$. They showed that the smallest cardinality of a locating-dominating code in $C_n(1,3)$ is at least $\lceil n/3 \rceil$ and at most $\lceil n/3 \rceil + 1$ for all $n \geq 9$. Moreover, they proved that the lower bound is strict when $n \equiv 0, 1, 4 \pmod{6}$ and conjectured that the lower bound can be increased by one for other $n$. In this paper, we prove their conjecture. Similarly, they showed that the smallest cardinality of an identifying code in $C_n(1,3)$ is at least $\lceil 4n/11 \rceil$ and at most $\lceil 4n/11 \rceil + 1$ for all $n \geq 11$. Furthermore, they proved that the lower bound is attained for most of the lengths $n$ and conjectured that in the rest of the cases the lower bound can improved by one. This conjecture is also proved in the paper. The proofs of the conjectures are based on a novel approach which, instead of making use of the local properties of the graphs as is usual to identification and location-domination, also manages to take advantage of the global properties of the codes and the underlying graphs.