On the complexity of graph coloring with additional local conditions

TL;DR

This paper proves that the semi-matching coloring problem, which involves specific local conditions on color classes, is NP-complete for any fixed number of colors greater than or equal to three, highlighting its computational complexity.

Contribution

It establishes NP-completeness of the semi-matching coloring problem and a variant with triangle color difference constraints for fixed k ≥ 3.

Findings

Semi-matching coloring problem is NP-complete for all fixed k ≥ 3.

A variant with triangle color difference constraints is also NP-complete.

Complexity results hold for fixed number of colors, emphasizing computational difficulty.

Abstract

Let $G = (V,E)$ be a finite simple graph. Recall that a proper coloring of $G$ is a mapping $\varphi: V\to\{1,\ldots,k\}$ such that every color class induces an independent set. Such a $\varphi$ is called a semi-matching coloring if the union of any two consecutive color classes induces a matching. We show that the semi-matching coloring problem is NP-complete for any fixed $k\geqslant 3$, and we get the same result for another version of this problem in which any triangle of G is required to have vertices whose colors differ at least by three.