On traceability of claw-o_{-1}-heavy graphs

TL;DR

This paper investigates conditions under which claw-$o_{-1}$-heavy graphs are guaranteed to contain a Hamilton path, focusing on additional restrictions involving forbidden subgraphs.

Contribution

It establishes new traceability results for claw-$o_{-1}$-heavy graphs under specific forbidden induced subgraph conditions.

Findings

Claw-$o_{-1}$-heavy graphs are traceable with certain additional constraints.

Identifies specific forbidden subgraphs that ensure traceability.

Provides new sufficient conditions for Hamilton path existence in these graphs.

Abstract

A graph is called traceable if it contains a Hamilton path, i.e., a path passing through all its vertices. Let $G$ be a graph on $n$ vertices. $G$ is called claw-$o_{-1}$-heavy if every induced claw ($K_{1,3}$) of $G$ has a pair of nonadjacent vertices with degree sum at least $n-1$ in $G$. In this paper we show that a claw-$o_{-1}$-heavy graph $G$ is traceable if we impose certain additional conditions on $G$ involving forbidden induced subgraphs.