Distant Set Distinguishing Total Colourings of Graphs

The Total Colouring Conjecture suggests that ∆ + 3 colours ought to suffice in order to provide a proper total colouring of every graph G with maximum degree ∆. Thus far this has been confirmed up to an additive constant factor, and the same holds even if one additionally requires every pair of neighbours in G to differ with respect to the sets of their incident colours, so called pallets. Within this paper we conjecture that an upper bound of the form ∆ + C, for a constant C > 0 still remains valid even after extending the distinction requirement to pallets associated with vertices at distance at most r, if only G has minimum degree δ larger than a constant dependent on r. We prove that such assumption on δ is then unavoidable and exploit the probabilistic method in order to provide two supporting results for the conjecture. Namely, we prove the upper bound (1 + o(1))∆ for every r, and show that for any fixed ∈ (0, 1] and r, the conjecture holds if δ > ε∆, i.e., in particular for regular graphs.