Matthews–Sumner Conjecture

Prove or disprove the Matthews–Sumner Conjecture that every 4-connected, claw-free graph is hamiltonian, equivalently traceable.

Background

The thesis discusses classical Hamiltonicity questions for highly connected claw-free graphs. It states the Matthews–Sumner conjecture and notes that, by a result cited there, its Hamiltonian formulation is equivalent to the assertion that every 4-connected claw-free graph is traceable.

References

Many papers have been written on hamiltonian cycles in $k$-connected, claw-free graphs for $k =2$ and $k =3$. Matthews and Sumner showed that a 3-connected, claw-free graph with fewer than 20 vertices is hamiltonian, and conjectured that if $G$ is a 4-connected, claw-free graph then $G$ is hamiltonian.

— Detours In Graphs  (2507.12086 - Bullock, 16 Jul 2025) in Chapter 3, Section "Introduction"

In addition, a notable unsolved problem is the Matthews–Sumner conjecture [27], which asks whether every 4-connected claw-free graph is Hamiltonian.

— On the properties of claw-free 1-planar graphs  (2501.15124 - Zhang et al., 25 Jan 2025) in Section 1.1, page 2

In addition, a notable unsolved problem is the Matthews–Sumner conjecture [27], which asks whether every 4-connected claw-free graph is Hamiltonian.

— On the properties of claw-free 1-planar graphs  (2501.15124 - Zhang et al., 25 Jan 2025) in Section 1.1, page 2

In addition, a notable unsolved problem is the Matthews–Sumner conjecture [27], which asks whether every 4-connected claw-free graph is Hamiltonian.

— On the properties of claw-free 1-planar graphs  (2501.15124 - Zhang et al., 25 Jan 2025) in Section 1.1, page 2