A vertex colouring of some graph is called perfect if each vertex of colour i has the same number aᵢⱼ of neighbours of colour j. Here we determine all perfect colourings of the edge graphs of the hypercube in dimensions 4 and 5 by two and three colours, respectively. For comparison we list all perfect colourings of the edge graphs of the simplex in dimensions 4 and 5, respectively.
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Dirk Frettlöh (2024) studied this question.
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