Rainbow cycles in flip graphs

TL;DR

This paper investigates the existence of r-rainbow cycles in various flip graphs related to geometric and non-geometric combinatorial objects, providing results on when such cycles exist or do not for different parameters.

Contribution

It extends the concept of rainbow cycles to multiple flip graph classes and determines their existence or non-existence across different parameters.

Findings

Proves existence of rainbow cycles in certain flip graphs.

Establishes non-existence of rainbow cycles in other cases.

Provides comprehensive results across five different flip graph settings.

Abstract

The flip graph of triangulations has as vertices all triangulations of a convex $n$-gon, and an edge between any two triangulations that differ in exactly one edge. An $r$-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly $r$ times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of $r$-rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex $n$-gon, the flip graph of plane trees on an arbitrary set of $n$ points, and the flip graph of non-crossing perfect matchings on a set of $n$ points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of $\{1,2,\dots,n\}$ and the flip graph of $k$-element subsets of $\{1,2,\dots,n\}$. In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of $r$, $n$ and~$k$.