Graphs with Flexible Labelings

TL;DR

This paper characterizes graphs that admit flexible labelings, allowing infinitely many realizations with fixed edge distances, through combinatorial edge colorings with cycle restrictions.

Contribution

It provides a new combinatorial characterization of graphs with flexible labelings based on edge colorings and cycle restrictions.

Findings

Characterization of graphs with flexible labelings via edge colorings.

Necessary and sufficient conditions for the existence of such colorings.

Insights into the structure of graphs allowing flexible realizations.

Abstract

For a flexible labeling of a graph, it is possible to construct infinitely many non-equivalent realizations keeping the distances of connected points constant. We give a combinatorial characterization of graphs that have flexible labelings. The characterization is based on colorings of the edges with restrictions on the cycles. Furthermore, we give necessary criteria and sufficient ones for the existence of such colorings.