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

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.

💡 Research Summary

The paper investigates Hamiltonian properties of claw‑free graphs and line graphs of 3‑hypergraphs under domination number constraints, extending several classical results and addressing a version of Thomassen’s line‑graph conjecture. The authors first recall that every 2‑connected claw‑free graph with domination number γ≤2 is Hamiltonian (Ageev, 1994) and that recent work has pushed the bound to γ≤3 for Hamilton‑connectedness in 3‑connected claw‑free graphs (Vrána, Zhan, Zhang, 2022). Their main contributions are twofold.

Theorem 1.5 states that any 3‑connected claw‑free graph G with γ(G)≤5 is Hamiltonian, except when the closure cl(G) belongs to a specific exceptional family P′. The family P′ consists of graphs obtained from the Petersen graph by attaching pendant edges or subdividing incident edges at each vertex, possibly together with edge‑domination number ≤5. The proof uses Ryjáček’s closure operation, which transforms a claw‑free graph into the line graph of a triangle‑free graph H. By examining the core co(H) (the graph obtained after suppressing degree‑2 vertices and removing pendant edges) and applying the Nine‑Point Theorem together with its extensions, the authors show that H contains a dominating closed trail covering a prescribed set of vertices. Harary‑Nash‑Williams’ correspondence then yields a Hamilton cycle in L(H)=cl(G). The only obstruction is when co(H) contracts to the Petersen graph, which precisely characterises the exceptional class P′. Hence the bound γ≤5 is best possible.

Theorem 1.6 strengthens the result to Hamilton‑connectedness: any 3‑connected claw‑free graph with γ≤4 is Hamilton‑connected unless its M‑closure (a strengthened closure preserving Hamilton‑connectedness) belongs to another exceptional family W′. The family W′ is derived from the Wagner graph by attaching pendant edges, adding double edges, subdividing edges, or inserting a double edge between a vertex and a subdivision vertex. Because ordinary closure does not preserve Hamilton‑connectedness, the authors employ the M‑closure introduced by Ryjáček and Vrána. They prove that cl_M(G) is again a line graph of a multigraph H, and that Hamilton‑connectedness of G is equivalent to Hamilton‑connectedness of cl_M(G). Using results on internally dominating trails (IDTs) and the correspondence between IDTs in H and Hamilton paths between any two vertices in L(H), together with the same core‑reduction and the strengthened cycle theorems (Liu et al., 2021), they establish the existence of an IDT for any pair of edges in H, unless H contracts to the Petersen graph, which again corresponds to the exceptional family W′. Consequently, γ≤4 is the optimal bound for Hamilton‑connectedness in the 3‑connected claw‑free setting.

The paper then turns to line graphs of 3‑hypergraphs. By a result of Li, Ozeki, Ryjáček and Vrána (2020), Thomassen’s conjecture is equivalent to the statement that every 4‑connected line graph of a 3‑hypergraph is Hamilton‑connected. Building on this equivalence, Theorem 1.8 proves that every 3‑connected line graph of a 3‑hypergraph with domination number at most 4 is Hamiltonian. The proof mirrors the previous arguments: the preimage H of the line graph is essentially 3‑edge‑connected; after taking its core, the authors apply the same dominating‑closed‑trail results (Theorems 2.6 and 2.7) to guarantee a closed trail covering all edges, which translates via Harary‑Nash‑Williams into a Hamilton cycle in the line graph. The domination bound excludes the Petersen‑contraction case, ensuring the result holds universally.

Throughout, the authors carefully delineate the exceptional families P′ and W′, demonstrating that they indeed violate the claimed Hamiltonian properties, thereby confirming the sharpness of the domination bounds. The work unifies several strands: domination‑based sufficient conditions, closure operations for claw‑free graphs, and the interplay between hypergraph line graphs and classic Hamiltonicity conjectures. It opens avenues for further research, such as extending the domination bound to 4‑connected graphs for Hamilton‑connectedness, or investigating analogous results for r‑uniform hypergraphs with r>3.