Relationship between the distinguishing index, minimum degree and maximum degree of graphs

Abstract

Let $δ$ and $Δ$ be the minimum and the maximum degree of the vertices of a simple connected graph $G$, respectively. The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. Motivated by a conjecture by Pilśniak (2017) that implies that for any $2$-connected graph $D'(G) \leq \lceil \sqrt{Δ(G)}\rceil +1$, we prove that for any graph $G$ with $δ\geq 2$, $D'(G) \leq \lceil \sqrt[δ]{Δ}\rceil +1$. Also, we show that the distinguishing index of $k$-regular graphs is at most $2$, for any $k\geq 5$.