For a graph $G$ and integer $k\geq1$, we define the token graph $F_k(G)$ to
be the graph with vertex set all $k$-subsets of $V(G)$, where two vertices are
adjacent in $F_k(G)$ whenever their symmetric difference is a pair of adjacent
vertices in $G$. Thus vertices of $F_k(G)$ correspond to configurations of $k$
indistinguishable tokens placed at distinct vertices of $G$, where two
configurations are adjacent whenever one configuration can be reached from the
other by moving one token along an edge from its current position to an
unoccupied vertex. This paper introduces token graphs and studies some of their
properties including: connectivity, diameter, cliques, chromatic number,
Hamiltonian paths, and Cartesian products of token graphs
This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are 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,such that each M i is compatible with M i+1 . This improves the previous best bound of k n − 2. We then study the conjecture: every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with n edges has an edge-disjoint compatible matching with approximately 4n/5 edges.
This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are 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 ∈ O(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 n − 2. We then study the conjecture: every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with n edges has an edge-disjoint compatible matching with approximately 4n/5 edges.
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