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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.
The theorem appears first in the 1891 article "Die Theorie der regulären graphs". To prove the theorem, Petersen's fundamental idea was to 'colour' the edges of a trail or a path alternatively red and blue, and then to use the edges of one or both colours for the construction of other paths or trials. [3]
His famous paper Die Theorie der regulären graphs [1] was a fundamental contribution to modern graph theory as we know it today. In 1898, he presented a counterexample to Tait's claimed theorem about 1-factorability of 3-regular graphs, which is nowadays known as the "Petersen graph". In cryptography and mathematical economics he made ...
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.
The 1-factorization of complete graphs is a special case of Baranyai's theorem concerning the 1-factorization of complete hypergraphs. One method for constructing a 1-factorization of a complete graph on an even number of vertices involves placing all but one of the vertices in a regular polygon , with the remaining vertex at the center.
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]
A subdivision of K 3,3 in the generalized Petersen graph G(9,2), showing that the graph is nonplanar.. In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski.