A Note on the Tattoo Index of Graphs

TL;DR

This paper introduces the tattoo index, an efficiency measure for graph tattooing processes, and presents initial results on its properties, enhancing understanding of graph cleaning dynamics.

Contribution

The paper proposes the new concept of the tattoo index for graphs, providing foundational results and insights into its properties.

Findings

Defined the tattoo index as an efficiency measure.

Established initial properties and bounds of the tattoo index.

Connected the tattoo index to graph cleaning processes.

Abstract

Consider a network $D$ of pipes which have to be cleaned using some cleaning agents, called brushes, assigned to some vertices. The tattooing of a simple connected directed graph $D$ is a particular type of the cleaning in which an arc are coloured by the colour of the colour-brush transiting it and the tattoo number of $D$ is a corresponding derivative of brush numbers in it. In this paper, we introduce a new concept, called the tattoo index of a given graph $G$, which is an efficiency index related to the tattooing sequence and we establish some introductory results on this parameter.