FAW 2019: Frontiers in Algorithmics pp 51-62

# Local Coloring: New Observations and New Reductions

• Jie You
• Yixin Cao
• Jianxin Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11458)

## 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 every set of 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 show that it cannot be solved in subexponential time, unless the Exponential Time Hypothesis fails, and a new reduction for the NP-hardness of this problem. Our structural observations behind these reductions are of independent interests. We close the paper with a short remark on local colorings of perfect graphs.

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