The locating chromatic number of generalized Petersen graphs with small order

Abstract

It was conjectured by Asmiati (2018) that the generalized Petersen graph $P\left(n,k\right)$ has a locating chromatic number 4 if and only if $(noddandk=1)$ or $(n=4andk=2)$. In this paper, we give a negative answer to the conjecture posed by Asmiati. As a consequence, we are able to exhibit many counterexamples to the recent conjecture proposed, by proving that if $(5\le n\le 12)$ and $(2\le k\le \lfloor \frac{n-1}{2}\rfloor )$ and $(n,k)\ne (12,5)$, then ${\chi }_{{}_L}\left(P(n,k)\right)=4$.