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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 ...
The friendship graphs (graphs formed by connecting a collection of triangles at a single common vertex) provide examples of graphs that are factor-critical but not Hamiltonian. If a graph G is factor-critical, then so is the Mycielskian of G. For instance, the Grötzsch graph, the Mycielskian of a five-vertex cycle-graph, is factor-critical. [4]
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]
In a cubic graph with a perfect matching, the edges that are not in the perfect matching form a 2-factor. By orienting the 2-factor, the edges of the perfect matching can be extended to paths of length three, say by taking the outward-oriented edges. This shows that every cubic, bridgeless graph decomposes into edge-disjoint paths of length ...
with a corresponding factor graph shown on the right. Observe that the factor graph has a cycle. If we merge (,) (,) into a single factor, the resulting factor graph will be a tree. This is an important distinction, as message passing algorithms are usually exact for trees, but only approximate for graphs with cycles.
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. For instance a 1-factorization is an edge coloring with the additional property that each vertex is ...
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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.