Distinguishing graphs by total colourings

Abstract

We introduce the total distinguishing number D (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 (G), which are defined for colourings of vertices and edges, respectively. We obtain a general sharp upper bound:We also introduce the total distinguishing chromatic number χ D (G) similarly defined for proper total colourings of a graph G. We prove that χ D (G) ≤ χ (G) + 1 for every connected graph G with the total chromatic number χ (G). Moreover, if χ (G) ≥ ∆(G)+ 2, then χ D (G) = χ (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.

Symmetry breaking indices for the Cartesian product of graphs

“…For example Collins and Trenk [5] introduced breaking symmetries of graphs via proper coloring, Fisher and Isaak [7] in 2008 used edge coloring to break symmetries of the Cartesian product of complete graphs, and in 2015 the distinguishing index D ′ (G)of a graph G is introduced by Kalinowski and Pilśniak [12] as the minimum number of colors required for a distinguishing edge coloring. Moreover, Kalinowski, Pilśniak and Woźniak, in [13] did a similar thing for the total coloring and defined the analogous D ′′ (G) to break symmetries of G by coloring its edges and vertices.…”

Distinguishing threshold of graphs

“…There are also several generalizations to the distinguishing coloring. Collins and Trenk [9] mixed the concept with proper coloring and introduced the distinguishing chromatic number χ D (G) of a graph G. Moreover, Kalinowski and Pilśniak [22] introduced the distinguishing index D ′ (G) and the distinguishing chromatic index χ ′ D (G), while they, along with Woźniak, in [23] did a similar thing for the total coloring and defined the analogous D ′′ (G) and χ ′′ D (G). The literature is also rich in results for product graphs.…”

Bounds for Distinguishing Invariants of Infinite Graphs

“…The most recent generalization of the distinguishing number was made by Kalinowski, Pilśniak and Woźniak in [8]. They introduced the total distinguishing number D (G) as the minimum number of colours needed to colour vertices and edges of a graph G such that this colouring is only preserved by the trivial automorphism.…”

“…It is clear by definition that D (G) max{D(G), D (G)}. It was proved in [8] that D (G) ∆(G) for a connected graph G with at least three vertices. Moreover, they introduced the total distinguishing chromatic number χ D (G) as the minimum number of colours needed to properly colour vertices and edges of G in the way that only the trivial automorphism preserves it.…”