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 $p\geq 3$.

翻译：我们证明,4和5套定期汉密尔顿图上的3套彩色图仍然是NP硬体,这加强了Dailey[Disc. Math.'80]和Fleischner和Sabidussi[J.图.Theor.'02]的结果;此外,我们证明,每套3套汉密尔顿平面图中每套6美元和每套3美元定购的1套汉密尔顿平面图中3套NP硬体。