1-11-representability of all 4-colourable graphs

Establish or disprove that every 4-colourable graph is 1-11-representable.

Background

The paper proves that all graphs on at most eight vertices are 1-11-representable and notes that a stronger result for all 4-colourable graphs would improve the bound in the multi-1-11-representation theorem from 24 to 32 vertices. The authors explicitly present the universal 4-colourable case as an unresolved direction.

References

Are all 4-colourable graphs 1-11-representable? If this is the case, the result of Theorem 4.2 could be immediately improved by replacing 24 with 32.

— 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