Rainbow Stackings of Random Edge-Colorings

TL;DR

This paper establishes a precise threshold for the existence of rainbow stackings in random edge-colored complete graphs, advancing understanding of color superimposition constraints in graph theory.

Contribution

It determines a sharp threshold for the parameter r that guarantees the existence or nonexistence of rainbow stackings in random edge-colorings.

Findings

Identifies a sharp threshold for r based on m and n.

Provides probabilistic bounds for rainbow stacking existence.

Enhances understanding of edge-coloring superimpositions in complete graphs.

Abstract

A rainbow stacking of $r$-edge-colorings $\chi_1, \ldots, \chi_m$ of the complete graph on $n$ vertices is a way of superimposing $\chi_1, \ldots, \chi_m$ so that no edges of the same color are superimposed on each other. We determine a sharp threshold for $r$ (as a function of $m$ and $n$) governing the existence and nonexistence of rainbow stackings of random $r$-edge-colorings $\chi_1,\ldots,\chi_m$.