Hardness of 4-Colourings G-Colourable Graphs

Sergey Avvakumov; Marek Filakovský; Jakub Opršal; Gianluca Tasinato; Uli Wagner

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arXiv · 2504.07592

Hardness of 4-Colourings G-Colourable Graphs

Sergey Avvakumov·Marek Filakovský·Jakub Opršal·Gianluca Tasinato·Uli Wagner

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Abstract

We study the complexity of a class of promise graph homomorphism problems. For a fixed graph H, the H-colouring problem is to decide whether a given graph has a homomorphism to H. By a result of Hell and Ne\v{s}et\v{r}il, this problem is NP-hard for any non-bipartite loop-less graph H. Brakensiek and Guruswami [SODA 2018] conjectured the hardness extends to promise graph homomorphism problems as follows: fix a pair of non-bipartite loop-less graphs G, H such that there is a homomorphism from G to H, it is NP-hard to distinguish between graphs that are G-colourable and those that are not H-colourable. We confirm this conjecture in the cases when both G and H are 4-colourable. This is a common generalisation of previous results of Khanna, Linial, and Safra [Comb. 20(3): 393-415 (2000)] and of Krokhin and Opr\v{s}al [FOCS 2019]. The result is obtained by combining the algebraic approach to promise constraint satisfaction with methods of topological combinatorics and equivariant obstruction theory.

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Sergey Avvakumov, Marek Filakovský, Jakub Opršal, Gianluca Tasinato, Uli Wagner. 2025-04-10. Hardness of 4-Colourings G-Colourable Graphs. https://arxiv.org/abs/2504.07592

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