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

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Abd Aziz Noor Alawiah; Jafari Rad Nader; Kamarulhaili Hailiza; Hasni Roslan

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Journal Paper

Paper Information

Journal: COMMUNICATIONS IN COMBINATORICS AND OPTIMIZATION Year:2024 | Volume:9 | Issue:1 Page(s): 37-49

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Title

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

Author(s)

Abd Aziz Noor Alawiah | Jafari Rad Nader | Kamarulhaili Hailiza | Hasni Roslan | Issue Writer Certificate

Keywords

Hamiltonian graphQ4

k-step Hamiltonian graphQ4

chromatic numberQ4

clique numberQ4

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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