The locating chromatic number of generalized Petersen graphs with small order

Abstract

It was conjectured by Asmiati (2018) that the generalized Petersen graph Pn,k 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≤n≤12) and (2≤k≤⌊n−12⌋) and (n,k)≠(12,5), then χLP(n,k)=4.