Hamilton paths and cycles in flip graphs of (almost-)perfect matchings

2026-07-06

AI Summary

This study investigates the Hamiltonicity of flip graphs of (near-)perfect matchings under combinatorial and geometric settings, where adjacency is defined via 2-flips or 1-flips. The authors systematically analyze the structure of these flip graphs on complete graphs, complete bipartite graphs, and geometric graphs induced by planar point sets. In the combinatorial setting, they provide the first complete characterization of Hamiltonian connectivity or laceability. In contrast, in the geometric setting, they prove that no Hamiltonian path or cycle exists—any such path necessarily omits an exponential number of vertices—yet they construct a cycle covering nearly all vertices when the points are in convex position. By integrating techniques from graph theory, combinatorics, and computational geometry, the work reveals fundamental structural differences between the two settings.

Abstract

We consider the set of matchings of a graph and a local change operation, called a flip, between them. In the combinatorial setting, the base graphs are either complete graphs or complete bipartite graphs, and in the geometric setting, the graphs are embedded on point sets in the plane, with the requirement that edges must be drawn as straight lines and must not cross. For base graphs with an even number of vertices, we consider perfect matchings, i.e., all vertices are matched, and for base graphs with an odd number of vertices, we consider almost-perfect matchings, i.e., all but one vertex of the graph are matched. A 2-flip between two perfect matchings exchanges two edges, and a 1-flip between two almost-perfect matchings exchanges one edge. The corresponding flip graph has the set of perfect or almost-perfect matchings as vertices, with pairs of them connected by an edge if they differ in a 2-flip or 1-flip, respectively. In this work, we provide a comprehensive picture of Hamiltonicity properties of these flip graphs. We prove that the flip graphs in the combinatorial setting are Hamilton-connected, i.e., they admit a Hamilton path between any two vertices, or, if the flip graphs are bipartite, we prove that they are Hamilton-laceable, i.e., they admit a Hamilton path between any two vertices from different partition classes. In the geometric setting, we prove that any path in them misses exponentially many vertices, in particular, they have no Hamilton paths or cycles. For points in convex position and almost-perfect matchings under 1-flips, we complement this by constructing a cycle in the flip graph that visits almost all vertices.

Problem

• Hamilton paths
• Hamilton cycles
• flip graphs
• perfect matchings
• geometric graphs

Innovation

• flip graphs
• Hamiltonicity
• perfect matchings
• geometric matchings
• Hamilton-laceable