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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{this http URL} {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.
[1] Yaroslav Shitov,et al. On the complexity of graph coloring with additional local conditions , 2017, Inf. Process. Lett..
[2] Zehui Shao,et al. NP-completeness of local colorings of graphs , 2018, Inf. Process. Lett..
[3] David S. Johnson,et al. Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..
[4] Gary Chartrand,et al. Local colorings of graphs , 2003 .