Augmenting Geometric Graphs with Matchings

在几何和组合场景中翻转奇匹配

Flipping odd matchings in geometric and combinatorial settings

研究奇匹配的重配置问题，提出组合场景中任意两奇匹配可相互重配置的图的多项式时间特征，证明相关翻转图直径线性及转换问题NP完全，且凸点集几何场景中该问题为FPT。

Counting perfect matchings of cubic graphs in the geometric dual

Lovász and Plummer conjectured, in the mid 1970's, that every cubic graph G with no cutedge has an exponential in |V(G)| number of perfect matchings. In this work we show that every cubic planar graph G whose geometric dual graph is a stack triangulation has at least 3 times the golden ratio to |V(G)|/72 distinct perfect matchings. Our work builds on a novel approach relating Lovász and Plummer's conjecture and the number of so called groundstates of the widely studied Ising model from statistical physics.

Compatible Geometric Matchings

This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are \emph{compatible} if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings $M$ and $M'$ of the same set of $n$ points, for some $k\in\Oh{\log n}$, there is a sequence of perfect matchings $M=M_0,M_1,...,M_k=M'$, such that each $M_i$ is compatible with $M_{i+1}$. This improves the previous best bound of $k\leq n-2$. We then study the conjecture: \emph{every perfect matching with an even number of edges has an edge-disjoint compatibl…

Characterization of co-blockers for simple perfect matchings in a convex geometric graph

Consider the complete convex geometric graph on $2m$ vertices, $CGG(2m)$, i.e., the set of all boundary edges and diagonals of a planar convex $2m$-gon $P$. In [C. Keller and M. Perles, On the Smallest Sets Blocking Simple Perfect Matchings in a Convex Geometric Graph], the smallest sets of edges that meet all the simple perfect matchings (SPMs) in $CGG(2m)$ (called "blockers") are characterized, and it is shown that all these sets are caterpillar graphs with a special structure, and that their total number is $m \cdot 2^{m-1}$. In this paper we characterize the co-blockers for SPMs in $CGG(2m…

A proof for Padberg's conjecture on rank of matching polytope

Padberg introduced a geometric notion of ranks for (mixed) integer rational polyhedrons and conjectured that the geometric rank of the matching polytope is one. In this work, we prove that this conjecture is true.

A Geometric Theory for Hypergraph Matching

We develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: 'space barriers' from convex geometry, and 'divisibility barriers' from arithmetic lattice-based constructions. To formulate precise results, we introduce the setting of simplicial complexes with minimum degree sequences, which is a generalisation of the usual minimum degree condition. We determine the essentially best possible minimum degree sequence for finding an almost perfect match…

Rainbow perfect matchings and Hamilton cycles in the random geometric graph

Given a graph on $n$ vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length $n$ visiting each vertex once and with pairwise different colours on the edges. Similarly (for even $n$) a rainbow perfect matching is a collection of $n/2$ independent edges with pairwise different colours. In this note we show that if we randomly colour the edges of a random geometric graph with sufficiently many colours, then a.a.s. the graph contains a rainbow perfect matching (rainbow Hamilton cycle) if and only if the minimum degree is at least $1$ (respectively, at leas…