2-Colorable Perfect Matching is NP-complete in 2-Connected 3-Regular Planar Graphs

TL;DR

This paper proves that determining a 2-colorable perfect matching is NP-complete in 2-connected 3-regular planar graphs, confirming a longstanding claim and extending the complexity results to k-colorable perfect matchings.

Contribution

It provides the first complete proof that 2-colorable perfect matching is NP-complete in 2-connected 3-regular planar graphs and extends NP-completeness to k-colorable perfect matchings for all k ≥ 2.

Findings

NP-completeness of 2-colorable perfect matching in 2-connected 3-regular planar graphs

Confirmation of Schaefer's 1978 claim with a formal proof

NP-completeness of k-colorable perfect matching for all fixed k ≥ 2

Abstract

The 2-colorable perfect matching problem asks whether a graph can be colored with two colors so that each node has exactly one neighbor with the same color as itself. We prove that this problem is NP-complete, even when restricted to 2-connected 3-regular planar graphs. In 1978, Schaefer proved that this problem is NP-complete in general graphs, and claimed without proof that the same result holds when restricted to 3-regular planar graphs. Thus we fill in the missing proof of this claim, while simultaneously strengthening to 2-connected graphs (which implies existence of a perfect matching). We also prove NP-completeness of $k$-colorable perfect matching, for any fixed $k \geq 2$.