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

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.

• The paper establishes that flip graphs of perfect and almost-perfect matchings are Hamilton-connected or Hamilton-laceable across various combinatorial configurations.
• It uses recursive constructions and a directed flip analysis to resolve longstanding open problems and strengthen previous Hamiltonicity results.
• In geometric settings, the study identifies inherent obstructions while devising near-complete cycle factors via ear decomposition methods.

Hamiltonicity in Flip Graphs of (Almost-)Perfect Matchings

Introduction and Background

The study of flip graphs formed from (almost-)perfect matchings, and the investigation of their Hamiltonicity properties, is a central topic at the interface of combinatorics, graph theory, and geometric computation. A flip graph is constructed by taking combinatorial or geometric matchings as vertices, with adjacency defined by minimal local operations ("flips")—namely, 2-flips for perfect matchings and 1-flips for almost-perfect matchings. The existence of Hamilton paths or cycles in such graphs, particularly under natural local transformations, has implications for Gray code generation, reconfiguration theory, and the study of associated polytopal skeleta.

This paper provides a comprehensive structural and algorithmic analysis of Hamiltonicity in flip graphs defined over both combinatorial (complete or complete bipartite graphs) and geometric (non-crossing matchings in the Euclidean plane) matchings, for both even and odd order. The results resolve open problems, significantly strengthen known results, and introduce new directed settings that reveal a more granular picture.

Main Results

Combinatorial Settings: Hamilton-Connectedness and Laceability

The paper establishes that, in the combinatorial case:

• The flip graph of perfect matchings in the complete graph $K_n$ ($n=2m$), under the adjacency induced by 2-flips, is Hamilton-connected: there exists a Hamilton path between any pair of vertices (Theorem~1). This sharply strengthens prior results that guaranteed only the existence of a Hamilton cycle (2607.04687).
• The flip graph of almost-perfect matchings in $K_{2m+1}$, with adjacency via 1-flips, is also Hamilton-connected (Theorem~2), affirmatively answering a problem posed by Aichholzer, Dorfer, Rieck, and Verciani. The construction is algorithmically efficient, supporting constant-average-time enumeration.
• For complete bipartite graphs ($K_{n,n}$ and $K_{n,n+1}$), the corresponding flip graphs of perfect or almost-perfect matchings are Hamilton-laceable: for any pair in opposite partition classes, there is a Hamilton path (Theorems~3, 4). These cases are closely related to Hamiltonicity in Cayley or Schreier graphs of the symmetric group under transpositions or star transpositions.
• The directed setting, where matching edges have orientations and flips respect directionality, reveals a taxonomy of flip types, some leading to disconnected, bipartite, or non-bipartite flip graphs. For combinations of flip types that yield connected graphs, explicit Hamilton-connectedness or laceability is established.

Geometric Setting: Obstructions to Hamiltonicity, Long Cycles

For non-crossing (almost-)perfect matchings on point sets in convex position:

• For perfect matchings on $n=2m$ points, Hernando, Hurtado, and Noy’s earlier result is restated and strengthened: a Hamilton cycle exists for even $m$; for odd $m$, any path misses $\Theta(2^m / m^{3/2})$ vertices, precluding Hamilton paths or cycles.
• For almost-perfect matchings on $n=2m+1$ points, the flip graphs are shown to lack both Hamilton paths and cycles—every path misses exponentially many vertices, $n=2m$0 for large $n=2m$1 (Theorem~5). Importantly, this is not due to bipartition imbalance but a prevalence of degree-2 vertices imposed by geometric constraints—a distinctly new obstruction.
• Despite exponential gaps in coverage, the authors construct cycles that traverse almost all vertices, specifically a $n=2m$2 fraction (Theorem~6). The cycle factor construction, followed by controlled gluing via substructures termed "ears," ensures that missing vertices are an asymptotically vanishing share.
• For points in general position, the lower bound can be tightened even further to $n=2m$3 missed vertices.

Methodological Innovations

• Recursive Hamilton Path Constructions: Inductive proofs, coupled with explicit path construction strategies, demonstrate Hamilton-connectedness in highly symmetric flip graphs and support efficient combinatorial enumeration algorithms.
• Directed Edge Flips and Fine-grained Analysis: By partitioning flips into fine types based on edge orientations, the work distinguishes which local operations suffice for full Hamiltonicity, revealing the subtle algebraic structure of these graphs.
• Ear Decomposition and Cycle Gluing: In geometric settings, cycle factors based on the rotation of matchings are systematically glued together using alternating cycles associated with "ears," permitting analytic control over the vertex set coverage and establishing long cycles.

Notable Numerical and Structural Results

• Hamilton-Connectedness for $n=2m$4: The flip graph on $n=2m$5 perfect matchings of $n=2m$6 with 2-flips is Hamilton-connected, while the analogous property for almost-perfect matchings under 1-flips resolves a standing open question (Section~3).
• Hamilton-Laceability in Bipartite Cases: All permutations (encoded as perfect matchings in $n=2m$7) can be traversed via a Hamilton path using only transpositions from a transposition tree (Section~3.3), drawing on deep links with the generation of permutations.
• Exponential Lower Bounds: In the geometric setting with odd $n=2m$8, any path misses at least $n=2m$9 matchings (Theorem~5).
• Almost-full Coverage: Even though no Hamilton path or cycle exists for almost-perfect geometric matchings with odd $K_{2m+1}$0, cycles can be constructed that visit at least $K_{2m+1}$1 fraction of vertices (Theorem~6).

Implications and Theoretical Significance

The results provide a unified framework for understanding Hamiltonicity in flip graphs of (almost-)perfect matchings, bridging algebraic, combinatorial, and geometric methodologies. The positive results not only settle open problems but also imply concrete algorithms for enumeration and traversal (combinatorial Gray codes) of matchings—supporting applications in combinatorial generation and random sampling.

The negative results in the geometric setting underscore the intrinsic limitations imposed by planarity and point-set structure, charting the boundaries of Gray code design in geometric contexts. The ear-based analysis of cycle factors introduces a robust toolkit for bounding coverage and constructing almost-spanning cycles, relevant for related polytopal complexes.

By introducing directed variants and analyzing their connectivity and Hamiltonicity, the paper opens new avenues for studying reconfiguration problems and flip graphs in more constrained or structured state spaces, aligning with the recent complexity-theoretic developments showing the hardness of flip-distance and reconfiguration computation.

Conclusion

This work delivers a comprehensive analysis of Hamilton paths and cycles in flip graphs of (almost-)perfect matchings, settling previously unresolved questions, introducing new combinatorial and algorithmic constructions, and establishing both the possibilities and fundamental obstructions to Hamiltonicity in a wide variety of settings. The fine-grained results, both positive and negative, will serve as a foundation for further studies in combinatorial reconfiguration, geometric generation, and the algorithmic study of flip graphs and their associated polytopality.