On the local metric dimension of $K_5$-free graphs

Let $G$ be a graph with order $n(G) \geq 5$, local metric dimension $\dim_l(G)$, and clique number $\omega(G)$. In this paper, we investigate the local metric dimension of $K_5$-free graphs and prove that $\dim_l(G) \leq \lfloor\frac{2}{3}n(G)\rfloor$ when $\omega(G) = 4$. As a consequence of this finding, along with previous publications, we establish that if $G$ is a $K_5$-free graph, then $\dim_l(G) \leq \lfloor\frac{2}{5}n(G)\rfloor$ when $\omega(G) = 2$, $\dim_l(G) \leq \lfloor\frac{1}{2}n(G)\rfloor$ when $\omega(G) = 3$, and $\dim_l(G) \leq \lfloor\frac{2}{3}n(G)\rfloor$ when $\omega(G) = 4$. Notably, these bounds are sharp for planar graphs. These results for graphs with a clique number less than or equal to 4 provide a positive answer to the conjecture stating that if $n(G) \geq \omega(G) + 1 \geq 4$, then $\dim_l(G) \leq \left( \frac{\omega(G) - 2}{\omega(G) - 1} \right)n(G)$.