Abstract
Let G=(V,E) be a graph and c:(V∪E)→C be a proper total colouring of G, where C is a set of colours. We call c inclusion-free if for each vertex, the set of colours appearing on the vertex and the incident edges is not a subset of the respective sets of its neighbours. With a probabilistic argument we show that the minimum number of colours for inclusion-free total colouring, denoted by χ⊂″(G), is bounded from above by Δ+150logΔ for any graph G with large enough maximum degree Δ. Then we prove that χ⊂″(G)≤6 for any subcubic graph G, which meets the bound for adjacent vertex distinguishing total colouring.
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