Conjecture for independent sets of unions of claws

Determine whether unions of claws satisfy the Holroyd–Talbot property for every r; in particular, prove or disprove the conjectured extension from kP_2 and kP_3 to unions of 3-claws.

Background

A k-claw is the complete bipartite graph K_{1,k}; thus K_{1,1}=P_2 and K_{1,2}=P_3. The survey suggests unions of 3-claws as a potentially more tractable next case for extending the cycle-method arguments used for unions of paths.

References

However, the authors suggest that unions of 3-claws also continue this sequence of graphs and might more easily succumb to a cycle-method-like symmetry argument --- a k-claw is the complete bipartite graph K_{1,k}, so K_{1,1}=P_2 and K_{1,2}=P_3.

— A Survey of the Holroyd-Talbot Conjecture  (2501.16144 - Hurlbert, 27 Jan 2025) in Section 3, discussion following Theorem 20