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

Abstract: 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) ).