Tattooing and the Tattoo Number of Graphs

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 [Formula: see text].