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A k-factor of a graph is a spanning k-regular subgraph, and a k-factorization partitions the edges of the graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is a proper edge coloring with k colors. A 2 ...
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]
Visualization of 6 as a perfect number Logarithmic graph of the number of digits of the largest known prime number by year, nearly all of which have been Mersenne primes Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory .
A 1-factorization is a partition of the edge set of the graph into three perfect matchings, or equivalently an edge coloring of the graph with three colors. Every biconnected n-vertex cubic graph has O(2 n/2) 1-factorizations, and the prism graphs have Ω(2 n/2) 1-factorizations. [3] The number of spanning trees of an n-gonal prism graph is ...
A number where some but not all prime factors have multiplicity above 1 is neither square-free nor squareful. The Liouville function λ(n) is 1 if Ω(n) is even, and is -1 if Ω(n) is odd. The Möbius function μ(n) is 0 if n is not square-free. Otherwise μ(n) is 1 if Ω(n) is even, and is −1 if Ω(n) is odd.
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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.
1 p.m. ET, Netflix. What's on the line: Each team has a lot to play for, even though their playoff spots are secure. The Chiefs can clinch the No. 1 seed in the AFC with a win. They lead the ...