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The theorem appears first in the 1891 article "Die Theorie der regulären graphs". [1] By today's standards Petersen's proof of the theorem is complicated. A series of simplifications of the proof culminated in the proofs by Frink (1926) and König (1936). In modern textbooks Petersen's theorem is covered as an application of Tutte's theorem.
Petersen graph as Kneser graph ,. The Petersen graph is the complement of the line graph of .It is also the Kneser graph,; this means that it has one vertex for each 2-element subset of a 5-element set, and two vertices are connected by an edge if and only if the corresponding 2-element subsets are disjoint from each other.
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’.
The Petersen family. K 6 is at the top of the illustration, K 3,3,1 is in the upper right, and the Petersen graph is at the bottom. The blue links indicate ΔY- or YΔ-transforms between graphs in the family. In graph theory, the Petersen family is a set of seven undirected graphs that includes the Petersen graph and the complete graph K 6.
According to Brooks' theorem every connected cubic graph other than the complete graph K 4 has a vertex coloring with at most three colors. Therefore, every connected cubic graph other than K 4 has an independent set of at least n/3 vertices, where n is the number of vertices in the graph: for instance, the largest color class in a 3-coloring has at least this many vertices.
The Petersen graph, being a snark, has a chromatic index of 4: its edges require four colors. All other generalized Petersen graphs have chromatic index 3. These are the only possibilities, by Vizing's theorem. [12] The generalized Petersen graph G(9, 2) is one of the few graphs known to have only one 3-edge-coloring. [13]
Graph structure theorem; Grinberg's theorem; Grötzsch's theorem; H. Hall-type theorems for hypergraphs; Hall's marriage theorem; ... Petersen's theorem; Planar ...
Equivalently (because the Petersen graph has maximum degree three) every snark has a subgraph that can be formed from the Petersen graph by subdividing some of its edges. This conjecture is a strengthened form of the four color theorem , because any graph containing the Petersen graph as a minor must be nonplanar.