Distinguishing graphs by total colourings

Abstract

We introduce the total distinguishing number D ʺ( G ) of a graph G as the least number d such that G has a total colouring (not necessarily proper) with d colours that is only preserved by the trivial automorphism. This is an analog to the notion of the distinguishing number D ( G ) , and the distinguishing index D ʹ( G ) , which are defined for colourings of vertices and edges, respectively. We obtain a general sharp upper bound: D ʺ( G ) ≤ ⌈√Δ ( G )⌉ for every connected graph G . We also introduce the total distinguishing chromatic number χ ʺ D ( G ) similarly defined for proper total colourings of a graph G . We prove that χ ʺ D ( G ) ≤ χ ʺ( G ) + 1 for every connected graph G with the total chromatic number χ ʺ( G ) . Moreover, if χ ʺ( G ) ≥ Δ ( G ) + 2 , then χ ʺ D ( G ) = χ ʺ( G ) . We prove these results by setting sharp upper bounds for the minimal number of colours in a proper total colouring such that each vertex has a distinct set of colour walks emanating from it.