Linearly ordered colourings of hypergraphs
Tamio-Vesa Nakajima, Stanislav \v{Z}ivn\'y

TL;DR
This paper studies linearly ordered colourings of hypergraphs, providing polynomial-time algorithms for certain cases and NP-hardness results for others, revealing structural differences based on hypergraph uniformity.
Contribution
It introduces new complexity results for LO colourings of hypergraphs, including polynomial algorithms and NP-hardness proofs, and explores algebraic properties related to these colourings.
Findings
Polynomial-time LO k-colouring for 3-uniform hypergraphs with LO 2-colouring.
NP-hardness of LO k-colouring for r-uniform hypergraphs with r ≥ k+2.
Relationships between polymorphism minions vary with hypergraph uniformity.
Abstract
A linearly ordered (LO) -colouring of an -uniform hypergraph assigns an integer from to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for , 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 -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 -colouring, one can find in polynomial time an LO -colouring with . Second, given an -uniform hypergraph that admits an LO -colouring, we establish NP-hardness of finding an LO -colouring for every…
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