# An upper bound on the distinguishing index of graphs with minimum degree at least two

## TL;DR

This paper establishes an upper bound on the distinguishing index of graphs with minimum degree at least two, confirming a conjecture for a broader class of graphs and providing specific examples where the bound is tight.

## Contribution

It proves a general upper bound on the distinguishing index for graphs with minimum degree at least two, extending Pilśniak's conjecture.

## Key findings

- Proved that for graphs with minimum degree at least two, D'(G) ≤ ⌈√Δ(G)⌉ + 1.
- Confirmed the conjecture for a broader class of graphs.
- Presented examples where D'(G) ≤ ⌈√Δ(G)⌉.

## Abstract

The distinguishing index of a simple 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. It was conjectured by Pil\'sniak (2015) that for any 2-connected graph $D'(G) \leq \lceil \sqrt{\Delta (G)}\rceil +1$. We prove a more general result for the distinguishing index of graphs with minimum degree at least two from which the conjecture follows. Also we present graphs $G$ for which $D'(G)\leq \lceil \sqrt{\Delta }\rceil$.