Colour-blind Can Distinguish Colour Pallets

Jakub Przybyło

doi:10.37236/3744

Abstract

Let $c:E\to\{1,\ldots,k\}$ be an edge colouring of a connected graph $G=(V,E)$. Each vertex $v$ is endowed with a naturally defined pallet under $c$, understood as the multiset of colours incident with $v$. If $\delta(G)\geq 2$, we obviously (for $k$ large enough) may colour the edges of $G$ so that adjacent vertices are distinguished by their pallets of colours. Suppose then that our coloured graph is examined by a person who is unable to name colours, but perceives if two object placed next to each other are coloured differently. Can we colour $G$ so that this individual can distinguish colour pallets of adjacent vertices? It is proved that if $\delta(G)$ is large enough, then it is possible using just colours 1, 2 and 3. This result is sharp and improves all earlier ones. It also constitutes a strengthening of a result by Addario-Berry, Aldred, Dalal and Reed (2005).

Highlights

Given vertex v ∈ V, we denote by d(v) the degree of v, and by N (v) the neighbourhood of v in G
A not necessarily proper edge colouring c : E → {1, 2, . . . , k} is called neighbour distinguishing if for every edge uv ∈ E, the multiset of colours incident with u is distinct from the multiset of colours incident with v
The problem of finding minimum k so that every graph without a K2 component admits a neighbour distinguishing edge colouring with k colours first arose

Summary

Introduction

The problem of finding minimum k so that every graph without a K2 component admits a neighbour distinguishing edge colouring with k colours first arose This was greatly improved by Addario-Berry et al [1], who showed that four colours are sufficient and provided the following refinement for graphs of sufficiently large minimum degree.

Results

Conclusion