  A $k$-coloring of a graph is an assignment of integers between $1$ and $k$ to
vertices in the graph such that the endpoints of each edge receive different
numbers. We study a local variation of the coloring problem, which imposes
further requirements on three vertices: We are not allowed to use two
consecutive numbers for a path on three vertices, or three consecutive numbers
for a cycle on three vertices. Given a graph $G$ and a positive integer $k$,
the local coloring problem asks for whether $G$ admits a local $k$-coloring. We
give a characterization of graphs admitting local $3$-coloring, which implies a
simple polynomial-time algorithm for it. Li et
al.~[\href{http://dx.doi.org/10.1016/j.ipl.2017.09.013} {Inf.~Proc.~Letters 130
(2018)}] recently showed it is NP-hard when $k$ is an odd number of at least
$5$, or $k = 4$. We show that it is NP-hard when $k$ is any fixed even number
at least $6$, thereby completing the complexity picture of this problem. We
close the paper with a short remark on local colorings of perfect graphs.
