A note on breaking small automorphisms in graphs

Abstract This paper brings together two concepts in the theory of graph colourings: edge or total colourings distinguishing adjacent vertices and those breaking symmetries of a graph. We introduce a class of automorphisms such that edge colourings breaking them are connected to edge colouring distinguishing neighbours by multisets or sums. We call an automorphism of a graph G small if there exists a vertex of G that is mapped into its neighbour. The small distinguishing index of G , denoted D s ′ ( G ) , is the least number of colours in an edge colouring of G such that there does not exist a small automorphism of G preserving this colouring. We prove that D s ′ ( G ) ≤ 3 for every graph G without K 2 as a component, thus supporting, in a sense, the 1-2-3 Conjecture of Karonski, Łuczak and Thomason. We also consider an analogous problem for total colourings in connection with the 1-2 Conjecture of Przybylo and Woźniak.