Colouring the 1-skeleton of d -dimensional triangulations

While every plane triangulation is colourable with three or four colours, Heawood showed that a plane triangulation is 3-colourable if and only if every vertex has even degree. In d ≥ 3 dimensions, however, every k ≥ d + 1 may occur as the chromatic number of some triangulation of 𝕊 d . As a first step, Joswig [14] structurally characterised which combinatorial triangulations of 𝕊 d have a ( d + 1 ) -colourable 1-skeleton. In the 20 years since Joswig’s result, no characterisations have been found for any k > d + 1 . In this paper, we structurally characterise which combinatorial triangulations of 𝕊 d have a ( d + 2 ) -colourable 1-skeleton: they are precisely the combinatorial triangulations that have a subdivision such that for every ( d - 2 ) -cell, the number of incident ( d - 1 ) -cells is divisible by three.