A Note on Construction of Dual-Hamiltonian Graphs

A connected simple graph is said dual-hamiltonian if its vertex set has a $2$-coloring such that each color class induces a tree. We call such a coloring a hamiltonian coloring. We prove that if $G$ is a graph with a certain type of hamiltonian coloring and $T$ is a tree, then $G\times T$ is also dual-hamiltonian having the same certain type of hamiltonian coloring. This result is used to constructed a class of dual-hamiltonian graphs, which includes the hypercubes and other multidimensional grids.