Local Coloring and its Complexity

TL;DR

This paper investigates the complexity of a local graph coloring variation, providing a complete classification of its computational difficulty for various fixed numbers of colors.

Contribution

It characterizes graphs with local 3-colorings, offers a polynomial-time algorithm for them, and proves NP-hardness for fixed even numbers at least 6, completing the complexity landscape.

Findings

Polynomial-time algorithm for local 3-coloring.

NP-hardness for fixed even k ≥ 6.

Completes the complexity classification for local coloring.

Abstract

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.