Linearly Ordered Colourings of Hypergraphs

A linearly ordered (LO) k-colouring of an r-uniform hypergraph assigns an integer from {1, ... , k } to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for r = 3, if two vertices in an edge are assigned the same colour, then the third vertex is assigned a larger colour (as opposed to a different colour, as in classic non-monochromatic colouring). Barto, Battistelli, and Berg [STACS’21] studied LO colourings on 3-uniform hypergraphs in the context of promise constraint satisfaction problems (PCSPs). We show two results. First, given a 3-uniform hypergraph that admits an LO 2-colouring, one can find in polynomial time an LO k-colouring with \( k=O(\sqrt [3]{n \log \log n / \log n} \) . Second, given an r-uniform hypergraph that admits an LO 2-colouring, we establish NP-hardness of finding an LO k-colouring for every constant uniformity r≥k+2. In fact, we determine relationships between polymorphism minions for all uniformities r≥ 3, which reveals a key difference between r< k+2 and r≥ k+2 and which may be of independent interest. Using the algebraic approach to PCSPs, we actually show a more general result establishing NP-hardness of finding an LO k-colouring for LO ℓ-colourable r-uniform hypergraphs for 2 ≤ ℓ ≤ k and r ≥ k - ℓ + 4.