On chromatic number and clique number in k-step Hamiltonian graphs

Document Type : Original paper

Authors

1 School of Mathematical Sciences , Universiti Sains Malaysia, 11800 Penang, Malaysia

2 Department of Mathematics, Shahed University, Tehran, Iran

3 Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu Malaysia

Abstract

A graph G of order n is called kstep Hamiltonian for k1 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,,n1. 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 kstep Hamiltonian graphs for k2. We present upper bounds for the chromatic number in kstep Hamiltonian graphs and give characterizations of graphs achieving the equality of the bounds. We also present an upper bound for the clique number in kstep Hamiltonian graphs and characterize graphs achieving equality of the bound.

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Main Subjects


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