Tattooing and the Tattoo Number of Graphs

TL;DR

This paper introduces the concept of tattooing in directed graphs, a novel variation of graph cleaning that involves coloring arcs with brushes and their blends, defining the tattoo number as a derivative of the brush number.

Contribution

It proposes a new graph parameter called the tattoo number, extending the brush number concept with coloring and blending of brushes in directed graphs.

Findings

Defined the tattooing process and tattoo number for directed graphs.

Established relationships between brush number and tattoo number.

Explored conditions for tattooing along out-arcs with color blends.

Abstract

Consider a network $D$ of pipes which have to be cleaned using some cleaning agents, called brushes, assigned to some vertices. The minimum number of brushes required for cleaning the network $D$ is called its brush number. 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. Tattooing along an out-arc of a vertex $v$ may proceed if a minimum set of colour-brushes is allocated (primary colours) or combined with those which have arrived (including colour blends) together with mutation of permissible new colour blends, has cardinality greater than or equal to $d^+_G(v)$.