On hamiltonian colorings of graphs Original Research Article

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 of image, where image is the length of a longest image–image path in image. For a hamiltonian coloring image, image is the largest color assigned to a vertex of image; while the hamiltonian chromatic number image over all hamiltonian colorings image of image. The circumference image of a graph image is the length of a longest cycle in image. A lower bound for image is given in terms of the number of vertices that receive colors between two specified colors in a hamiltonian coloring of image. As a consequence of this result, it follows that if there exists a hamiltonian coloring of a connected graph image of order image such that at least image vertices of image are colored the same, then image is hamiltonian. Also, if there exists a hamiltonian coloring of a connected graph image of order image such that at least image vertices of image are colored with one of two consecutive colors, then image. Furthermore, it is shown that if image is a connected graph of order image with image, then image. Moreover, if image is a connected graph of order image that is not a star, then image.