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

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

Abstract

A graph $G$ of order $n$ is called $k-$step Hamiltonian for $k\geq 1$ if we can label the vertices of $G$ as $v_1,v_2,\ldots,v_n$ such that $d(v_n,v_1)=d(v_i,v_{i+1})=k$ for $i=1,2,\ldots,n-1$. 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 $k\geq 2$. We present upper bounds for the chromatic 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.

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References

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Communications in Combinatorics and Optimization
Volume 9, Issue 1
March 2024
Pages 37-49

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