On hamiltonian colorings for some graphs

For a connected graph image and any two vertices image and image in image, let image denote the length of a longest image path in image. A hamiltonian coloring of a connected graph image of order image is an assignment image of colors (positive integers) to the vertices of image such that image for every two distinct vertices image and image in image. The value image of a hamiltonian coloring image is the maximum color assigned to a vertex of image. The hamiltonian chromatic number image of image is image taken over all hamiltonian colorings image of image. In this paper we discuss the hamiltonian chromatic number of graphs image with image. As examples, we determine the hamiltonian chromatic number for a class of caterpillars, and double stars.