on chromatic number and clique number in k-step hamiltonian graphs

A graph g of order n is called k−step hamiltonian for k ≥ 1 if we can label the vertices of g as v1, v2, . . . , vn such that d(vn, v1) = d(vi, vi+1) = k for i = 1, 2, . . . , n−1. the (vertex) chromatic number of a graph g is the minimum number of colors needed to color the vertices of g so that no pair of adjacent vertices receive the same color. the clique number of g is the maximum cardinality of a set of pairwise adjacent vertices in g. in this paper, we study the chromatic number and the clique number in k−step hamiltonian graphs for k ≥ 2. we present upper bounds for thechromatic number in k−step hamiltonian graphs and give characterizations of graphs achieving the equality of the bounds. we also present an upper bound for the clique number in k−step hamiltonian graphs and characterize graphs achieving equality of the bound.