Rainbow Separating Path Systems

TL;DR

This paper introduces a colorful variant of separating path systems, analyzing their minimal sizes across different graph classes and color counts, revealing diverse asymptotic behaviors.

Contribution

It defines a new colorful separating path system concept and characterizes its minimal sizes and asymptotic behaviors for various graph classes.

Findings

Calculated minimum sizes for various graphs and color counts.

Identified three asymptotic behaviors as colors increase.

Provided examples illustrating each asymptotic behavior.

Abstract

We introduce a colorful version of separating path systems, in which two edges can only be separated from each other by two paths of distinct colors. We calculate the minimum sizes of such systems for various standard classes of graphs and numbers of colors. With respect to this setup, we identify three possible asymptotic behaviors for a class of graphs as the number of colors goes to infinity, and we find a wide range of examples that display each of these behaviors.