Hamiltonian Properties of 3-Connected Claw-Free Graphs and Line Graphs of 3-Hypergraphs

TL;DR

This paper establishes new upper bounds on domination numbers that guarantee Hamiltonian and Hamilton-connected properties in 3-connected claw-free graphs and line graphs of 3-hypergraphs, extending previous results.

Contribution

It extends existing research by determining the best possible domination number bounds for Hamiltonicity and Hamilton-connectivity in 3-connected claw-free graphs and line graphs of 3-hypergraphs.

Findings

3-connected claw-free graphs with domination number ≤ 5 are Hamiltonian (except some cases)

3-connected claw-free graphs with domination number ≤ 4 are Hamilton-connected (except some cases)

Line graphs of 3-hypergraphs with domination number ≤ 4 are Hamiltonian

Abstract

Motivated by Thomassen's well-known line graph conjecture, many researchers have explored sufficient conditions for claw-free graphs to be Hamiltonian or Hamilton-connected. In 1994, Ageev proved that every $2$-connected claw-free graph with domination number at most $2$ is Hamiltonian. In this paper, we extend this line of research to $3$-connected graphs by establishing the best possible upper bound on the domination number that guarantees Hamiltonicity. Specifically, we show that, except for some well-defined exceptional graphs, every $3$-connected claw-free graph $G$ with domination number at most $5$ is Hamiltonian. Furthermore, we prove that, apart from a few exceptional cases, every $3$-connected claw-free graph $G$ with domination number at most $4$ is Hamilton-connected, thereby generalizing earlier results of Zheng, Broersma, Wang and Zhang and Vr\'ana, Zhan and Zhang. We further investigate the Hamiltonian properties of line graphs of $3$-hypergraphs, and prove that every 3-connected line graph of a 3-hypergraph with domination number at most $4$ is Hamiltonian.