Finiteness of 6-connected claw-free 1-planar graphs

Determine whether only finitely many 6-connected claw-free 1-planar graphs exist.

Background

The paper gives two 6-connected claw-free 1-planar graphs and notes that the complete multipartite graph K2,2,2,2 is another example. However, the authors state that they have not discovered an infinite family of such graphs. Motivated by an analogy with the connectivity result for claw-free planar graphs, they explicitly ask whether the entire class of 6-connected claw-free 1-planar graphs is finite.

References

While the 1-planar graph K2,2,2,2 [17] and two 1-planar graphs in Fig. 2 are 6-connected and claw-free, we have not yet discovered a class of infinitely many 6-connected claw-free 1-planar graphs. On the other hand, in light of a potential parallel with Theorem 1.2, we propose the following problem. Problem 4.1. Does there exist only a finite number of 6-connected claw-free 1-planar graphs?

— On the properties of claw-free 1-planar graphs  (2501.15124 - Zhang et al., 25 Jan 2025) in Concluding remarks and open problems, Problem 4.1, page 10