Intertwining local (adjacency) metric dimension with the clique number of a graph

TL;DR

This paper establishes a relationship between local adjacency metric dimension and clique number in graphs, confirming a conjecture that provides an upper bound for the local metric dimension, with many graphs achieving equality.

Contribution

The paper proves an upper bound for the local adjacency metric dimension based on the clique number and confirms a conjecture relating it to the local metric dimension.

Findings

Upper bound for ${ m dim}_{A,l}(G)$ in terms of clique number and order

Confirmation of the conjecture linking ${ m dim}_l(G)$ to the upper bound

Existence of infinitely many graphs satisfying the equality

Abstract

Let $G$ be a simple connected graph with order $ n(G)$, local metric dimension $ {\rm dim}_l(G)$, local adjacency metric dimension $ {\rm dim}_{A,l}(G)$, and clique number $ \omega(G)$, where $G\not\cong K_{n(G)}$ and $\omega(G)\geq3$. It is proved that $ {\rm dim}_{A,l}(G) \leq \left\lfloor \left(\frac{\omega(G) - 2}{\omega(G) - 1}\right)n(G)\right\rfloor$. Consequently, the conjecture asserting that the latter expression is an upper bound for ${\rm dim}_l(G)$ is confirmed. It is important to note that there are infinitely many graphs that satisfy the equalities.