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

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 Vrána, 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.

Figures (5)

• Figure 1: The Petersen graph $P$ and the Wagner graph $W$.
• Figure 2: Illustrative examples for the construction in Definition \ref{['Defi1']}.
• Figure 3: The diamond $T_1$, the multitriangle $T_2$ and the triple edge $T_3$.
• Figure 4: Non-essentially edge cut in $IG'(H)$.
• Figure 5: The graphs $K_{1,2,3}$ and $K_{1,1,3}$.

Theorems & Definitions (19)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 2.1
• Theorem 2.2