Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs

Marek Filakovský; Tamio-Vesa Nakajima; Jakub Opršal; Gianluca Tasinato; Uli Wagner

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arXiv · 2312.12981

Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs

Marek Filakovský·Tamio-Vesa Nakajima·Jakub Opršal·Gianluca Tasinato·Uli Wagner

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Abstract

A linearly ordered (LO) $k$-colouring of a hypergraph is a colouring of its vertices with colours $1, \dots, k$ such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO $k$-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the `linearly ordered chromatic number' of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO $3$-colourable, and the case that it is not even LO $4$-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023).

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Marek Filakovský, Tamio-Vesa Nakajima, Jakub Opršal, Gianluca Tasinato, Uli Wagner. 2023-12-20. Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs. https://arxiv.org/abs/2312.12981

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