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In the mathematical discipline of graph theory, the 2-factor theorem, discovered by Julius Petersen, is one of the earliest works in graph theory. It can be stated as follows: [ 1 ] Let G {\displaystyle G} be a regular graph whose degree is an even number, 2 k {\displaystyle 2k} .
1-factorization of the Desargues graph: each color class is a 1-factor. The Petersen graph can be partitioned into a 1-factor (red) and a 2-factor (blue). However, the graph is not 1-factorable. In graph theory, a factor of a graph G is a spanning subgraph, i.e., a subgraph that has the same vertex set as G.
A factor graph is a bipartite graph representing the factorization of a function. In probability theory and its applications, factor graphs are used to represent factorization of a probability distribution function , enabling efficient computations, such as the computation of marginal distributions through the sum–product algorithm .
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 ...
In graph theory, two of Petersen's most famous contributions are: the Petersen graph, exhibited in 1898, served as a counterexample to Tait's ‘theorem’ on the 4-colour problem: a bridgeless 3-regular graph is factorable into three 1-factors and the theorem: ‘a connected 3-regular graph with at most two leaves contains a 1-factor’.
Example graph that has a vertex cover comprising 2 vertices (bottom), but none with fewer. In graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. In computer science, the problem of finding a minimum vertex cover is a classical optimization problem.
Another important factor of common development of graph theory and topology came from the use of the techniques of modern algebra. The first example of such a use comes from the work of the physicist Gustav Kirchhoff , who published in 1845 his Kirchhoff's circuit laws for calculating the voltage and current in electric circuits .
A factor-critical graph, together with perfect matchings of the subgraphs formed by removing one of its vertices. In graph theory, a mathematical discipline, a factor-critical graph (or hypomatchable graph [1] [2]) is a graph with n vertices in which every induced subgraph of n − 1 vertices has a perfect matching. (A perfect matching in a ...