3-Coloring on Regular, Planar, and Ordered Hamiltonian Graphs

TL;DR

This paper establishes that the computationally hard problem of 3-Coloring remains NP-hard on various classes of regular, planar, and Hamiltonian graphs, extending previous hardness results to broader graph families.

Contribution

It proves NP-hardness of 3-Coloring on 4- and 5-regular planar Hamiltonian graphs and on p-regular and p-ordered regular Hamiltonian graphs for specified p values, broadening known complexity boundaries.

Findings

3-Coloring is NP-hard on 4- and 5-regular planar Hamiltonian graphs.

NP-hardness extends to p-regular Hamiltonian graphs for p ≥ 6.

NP-hardness also holds for p-ordered regular Hamiltonian graphs for p ≥ 3.

Abstract

We prove that 3-Coloring remains NP-hard on 4- and 5-regular planar Hamiltonian graphs, strengthening the results of Dailey [Disc. Math.'80] and Fleischner and Sabidussi [J. Graph. Theor.'02]. Moreover, we prove that 3-Coloring remains NP-hard on $p$-regular Hamiltonian graphs for every $p\geq 6$ and $p$-ordered regular Hamiltonian graphs for every $p\geq 3$.