On a graph colouring problem

A graph G = (V, E) is said to be k-emulsive if it admits an edge-colouring ψ : E → [k] = {1, 2, …, k} such that, for any vertex-colouring ϕ : V → [k] there exists an edge e = {x, y} such that ϕ(x) = ϕ(y) = ψ(e). We show, by construction, that the complete graph on (1 + o(1))k2 vertices is k-emulsive. This settles a question raised by Cochand and Duchet.