Locally identifying colourings for graphs with given maximum degree

TL;DR

This paper introduces a new upper bound on the number of colours needed for locally identifying colourings in graphs with a given maximum degree, advancing understanding of graph colourings with local identification properties.

Contribution

The paper proves that any graph with maximum degree Δ admits a locally identifying colouring with at most 2Δ^2 - 3Δ + 3 colours, answering a previously open question.

Findings

Established an upper bound of 2Δ^2 - 3Δ + 3 colours for locally identifying colourings.

Extended results to locally identifying colourings with all distinct colours in each neighbourhood.

Applied the method to chordal graphs, demonstrating broader applicability.

Abstract

A proper vertex-colouring of a graph G is said to be locally identifying if for any pair u,v of adjacent vertices with distinct closed neighbourhoods, the sets of colours in the closed neighbourhoods of u and v are different. We show that any graph G has a locally identifying colouring with $2\Delta^2-3\Delta+3$ colours, where $\Delta$ is the maximum degree of G, answering in a positive way a question asked by Esperet et al. We also provide similar results for locally identifying colourings which have the property that the colours in the neighbourhood of each vertex are all different and apply our method to the class of chordal graphs.