Abstract :
Given a graph G = (V, E) having maximum degree δ with a proper vertex-colouring ϕ : V → {1,2 …, Δ + 2} there does not always exist a total colouring ψ : V ∪ E → {1, 2, …, Δ + 2} such that ψ(v) = ψ(v) for all vertices v ∈ V. We present an example with Δ = 5, which is lower than the maximum degree of any previous example.
