Chromatic Number and Hamiltonicity of Graphs

TL;DR

This paper explores the relationship between the chromatic number and Hamiltonian properties of graphs, establishing conditions under which a graph is Hamiltonian or has a specific structure based on its chromatic number.

Contribution

It provides a new criterion linking high chromatic number and connectivity to Hamiltonicity or a particular graph structure.

Findings

If $ ext{chromatic number} \, ext{chi}(G) \, ext{is at least} \, n - k$, then G is Hamiltonian or has a specific structure.

The result applies to $k$-connected graphs with $k \, ext{at least} \, 2$ and order $n \, ext{at least} \, 2k+1$.

The paper characterizes graphs with high chromatic number in terms of Hamiltonicity or a specific join structure.

Abstract

Let $G$ be a $k$ - connected ($k \geq 2$) graph of order $n$. If $\chi(G) \geq n - k$, then $G$ is Hamiltonian or $K_k \vee (K_k^c \cup K_{n - 2k})$ with $n \geq 2 k + 1$, where $\chi(G)$ is the chromatic number of the graph $G$.