# Is there an MMSNP formula for 3-colouring?

By MMSNP, I mean Monotone Monadic SNP without inequality. For $$k\in\mathbb{N}$$, the problem $$k$$-COLOURABILITY takes a graph $$G$$ as input and asks whether $$G$$ is $$k$$-colourable. It is well-known that for every $$k\in\mathbb{N}$$, $$k$$-COLOURABILITY is a finite-domain Constraint Satisfaction Problem (CSP). Does this suggest that there is an MMSNP formula for $$k$$-COLOURABILITY?

Is there an MMSNP formula for 3-COLOURABILITY?

It seems to me that the monotone condition makes it difficult to express that $$\{V_1,V_2,V_3\}$$ is a partition of the vertex set.