On the local metric dimension of $K_4$-free graphs

Ali Ghalavand; Sandi Klavžar; Xueliang Li

On the local metric dimension of $K_4$-free graphs

Combinatorics 2025-06-03 v1

Authors: Ali Ghalavand , Sandi Klavžar , Xueliang Li

Abstract

Let $G$ be a graph of order $n(G)$ , local metric dimension $\dim_l(G)$ , and clique number $\omega(G)$ . It has been conjectured that if $n(G) \geq \omega(G) + 1 \geq 4$ , then $\dim_l(G) \leq \left( \frac{\omega(G) - 2}{\omega(G) - 1} \right) n(G)$ . In this paper the conjecture is confirmed for the case $\omega(G) = 3$ . Consequently, a problem regarding the local metric dimension of planar graphs is also resolved.

Keywords

local metric dimension K_4-free graphs clique number planar graphs graph theory

Cite

@article{arxiv.2506.00414,
  title  = {On the local metric dimension of $K_4$-free graphs},
  author = {Ali Ghalavand and Sandi Klavžar and Xueliang Li},
  journal= {arXiv preprint arXiv:2506.00414},
  year   = {2025}
}

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