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A perfect 1-factorization (P1F) of a graph is a 1-factorization having the property that every pair of 1-factors is a perfect pair. A perfect 1-factorization should not be confused with a perfect matching (also called a 1-factor). In 1964, Anton Kotzig conjectured that every complete graph K 2n where n ≥ 2 has a perfect 1-factorization. So ...
A perfect matching is also called a 1-factor; see Graph factorization for an explanation of this term. In some literature, the term complete matching is used. Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs: [1]
An example showing how the FKT algorithm finds a Pfaffian orientation. Compute a planar embedding of G. Compute a spanning tree T 1 of the input graph G. Give an arbitrary orientation to each edge in G that is also in T 1. Use the planar embedding to create an (undirected) graph T 2 with the same vertex set as the dual graph of G.
The 1-factorization conjecture that if is odd or even and , respectively, then a -regular graph with vertices is 1-factorable. The perfect 1-factorization conjecture that every complete graph on an even number of vertices admits a perfect 1-factorization.
A chain graph is a graph which may have both directed and undirected edges, but without any directed cycles (i.e. if we start at any vertex and move along the graph respecting the directions of any arrows, we cannot return to the vertex we started from if we have passed an arrow). Both directed acyclic graphs and undirected graphs are special ...
Three people were shot and one person was stabbed following what authorities believe to be a family dispute at a restaurant in the Phoenix Sky Harbor International Airport.
For one, the research combines multiple holidays and data sources, but these sources aren’t all equal, Harkavy-Friedman said — some countries have 20 to 40 years’ worth of data, while the ...
In particular, a 1-factor is the same thing as a perfect matching. A factor-critical graph is a graph for which deleting any one vertex produces a graph with a 1-factor. factorization A graph factorization is a partition of the edges of the graph into factors; a k-factorization is a partition into k-factors.