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Arc-distinguishing of orientations of graphs

Aleksandra Gorzkowska, Jakub Kwaśny

Abstract

A distinguishing index of a (di)graph is the minimum number of colours in an edge (or arc) colouring such that the identity is the only automorphism that preserves that colouring. We investigate the minimum and maximum value of the distinguishing index over all orientations of a given graph $G$. We present sharp results for these parameters in terms of the distinguishing index of $G$ for trees, unbalanced bipartite graphs, traceable graphs and claw-free graphs. With this, we answer the question of Meslem and Sopena.

Arc-distinguishing of orientations of graphs

Abstract

A distinguishing index of a (di)graph is the minimum number of colours in an edge (or arc) colouring such that the identity is the only automorphism that preserves that colouring. We investigate the minimum and maximum value of the distinguishing index over all orientations of a given graph . We present sharp results for these parameters in terms of the distinguishing index of for trees, unbalanced bipartite graphs, traceable graphs and claw-free graphs. With this, we answer the question of Meslem and Sopena.
Paper Structure (4 sections, 12 theorems, 2 equations)

This paper contains 4 sections, 12 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Bipartite graphs
  3. Graphs with $D'(G)=2$
  4. Conclusions

Key Result

Theorem 2

MeslemSopena For every two integers $m$ and $n$, $2 \leq m < n$, the following hold:

Theorems & Definitions (20)

  • Theorem 2
  • Lemma 3
  • proof
  • Corollary 4
  • proof
  • Theorem 5
  • Corollary 6
  • Corollary 7
  • Theorem 8
  • proof
  • ...and 10 more