  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 $p\geq 3$.
