A Note on Construction of Dual-Hamiltonian Graphs

TL;DR

This paper investigates dual-hamiltonian graphs, showing how certain colorings are preserved under graph products, and constructs new classes of such graphs including hypercubes and grids.

Contribution

It proves that specific hamiltonian colorings are maintained under graph products with trees, enabling the construction of new dual-hamiltonian graphs.

Findings

Dual-hamiltonian property preserved under certain graph products

Construction of new dual-hamiltonian graphs including hypercubes

Identification of conditions for hamiltonian colorings in graph products

Abstract

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.