On the number of heterochromatic trees in nice and beautiful colourings of complete graphs

TL;DR

This paper investigates the quantity of heterochromatic spanning trees in specially classified edge-colourings of complete graphs, revealing quadratic and exponential growth depending on the coloring type.

Contribution

It introduces the concepts of nice and beautiful colourings and establishes lower bounds on heterochromatic trees in these classes, advancing understanding of graph colourings.

Findings

Nice colourings guarantee at least a quadratic number of heterochromatic trees.

Beautiful colourings ensure an exponential number of heterochromatic trees.

The results provide new bounds on heterochromatic spanning trees in complete graphs.

Abstract

We introduce classes of edge-colourings of the complete graph -- that we call nice and beautiful -- and study how many heterochromatic spanning trees appear under such colourings. We prove that if the colouring is nice, there is at least a quadratic number of different heterochromatic trees; and if the colouring is beautiful there is an exponential number of different such trees.