1-11-representability of all planar graphs

Establish or disprove that every planar graph is 1-11-representable.

Background

Planar graphs form a subclass of 4-colourable graphs. The authors propose their 1-11-representability as a more focused alternative if resolving the corresponding question for all 4-colourable graphs proves too difficult.

References

If proving or disproving that all 4-colourable graphs are 1-11-representable proves too challenging, one could instead address the same question for all planar graphs, a subclass of 4-colourable graphs.

— On 1-11-representability and multi-1-11-representability of graphs  (2501.13871 - Alshammari et al., 23 Jan 2025) in Section 5, Concluding remarks, page 9