Degree and neighborhood intersection conditions restricted to induced subgraphs ensuring Hamiltonicity of graphs

TL;DR

This paper establishes new Hamiltonicity conditions for graphs based on degree and neighborhood intersection restrictions within specific induced subgraphs, extending previous theoretical results.

Contribution

It introduces novel Hamiltonian criteria involving o-heavy and 1-heavy conditions combined with Fan-type degree or neighborhood intersection constraints.

Findings

2-connected o-heavy graphs are Hamiltonian under certain conditions

3-connected 1-heavy graphs are Hamiltonian with specific restrictions

Extends and improves previous Hamiltonicity results in graph theory

Abstract

Let claw be the graph $K_{1,3}$. A graph $G$ on $n\geq 3$ vertices is called \emph{o}-heavy if each induced claw of $G$ has a pair of end-vertices with degree sum at least $n$, and 1-heavy if at least one end-vertex of each induced claw of $G$ has degree at least $n/2$. In this note, we show that every 2-connected $o$-heavy or 3-connected 1-heavy graph is Hamiltonian if we restrict Fan-type degree condition or neighborhood intersection condition to certain pairs of vertices in some small induced subgraphs of the graph. Our results improve or extend previous results of Broersma et al., Chen et al., Fan, Goodman & Hedetniemi, Gould & Jacobson, and Shi on the existence of Hamilton cycles in graphs.