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Breaking small automorphisms of graphs of arbitrary cardinality

Marcin Stawiski

Abstract

We say that an edge colouring $c$ of a graph preserves an automorphism $\varphi$ if $\varphi$ maps each edge to an edge of the same colour. Otherwise, we say that $c$ breaks $\varphi$. We call an automorphism of a graph small if it moves some vertex to its neighbour. We study the edge colourings of graphs that break every small automorphism. Kalinowski, Pilśniak, and Woźniak proved that three colours are enough for such a colouring to exist for every finite graph without isolated edges. They conjectured that two colours are enough for every finite connected graph on at least six vertices. We confirm this conjecture in its more general version, namely for connected finite and infinite graphs of arbitrary cardinality.

Breaking small automorphisms of graphs of arbitrary cardinality

Abstract

We say that an edge colouring of a graph preserves an automorphism if maps each edge to an edge of the same colour. Otherwise, we say that breaks . We call an automorphism of a graph small if it moves some vertex to its neighbour. We study the edge colourings of graphs that break every small automorphism. Kalinowski, Pilśniak, and Woźniak proved that three colours are enough for such a colouring to exist for every finite graph without isolated edges. They conjectured that two colours are enough for every finite connected graph on at least six vertices. We confirm this conjecture in its more general version, namely for connected finite and infinite graphs of arbitrary cardinality.
Paper Structure (2 sections, 3 theorems)

This paper contains 2 sections, 3 theorems.

Table of Contents

  1. Introduction
  2. Main theorem

Key Result

Theorem 1

Let $G$ be a finite or infinite graph of order at least $6$. Then $D'_s(G)\leq 2$.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2: Kwaśny, Stawiski, 2025+ KS:regular
  • Theorem 3: Stawiski, Wilson 2024 Wilson
  • proof