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

Let $G$ be a graph with order $n(G)\ge 5$, 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 ${\dim }_l(G)\le \lfloor \frac{2}{3}n(G)\rfloor$ 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 ${\dim }_l(G)\le \lfloor \frac{2}{5}n(G)\rfloor$ when $\omega (G)=2$, ${\dim }_l(G)\le \lfloor \frac{1}{2}n(G)\rfloor$ when $\omega (G)=3$, and ${\dim }_l(G)\le \lfloor \frac{2}{3}n(G)\rfloor$ 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 $n(G)\ge \omega (G)+1\ge 4$, then ${\dim }_l(G)\le \left(\frac{\omega (G)-2}{\omega (G)-1}\right)n(G)$.