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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.
Petersen's theorem can also be applied to show that every maximal planar graph can be decomposed into a set of edge-disjoint paths of length three. In this case, the dual graph is cubic and bridgeless, so by Petersen's theorem it has a matching, which corresponds in the original graph to a pairing of adjacent triangle faces. Each pair of ...
The Kneser graph K(n, 1) is the complete graph on n vertices. The Kneser graph K(n, 2) is the complement of the line graph of the complete graph on n vertices. The Kneser graph K(2n − 1, n − 1) is the odd graph O n; in particular O 3 = K(5, 2) is the Petersen graph (see top right figure). The Kneser graph O 4 = K(7, 3), visualized on the right.
Coxeter's notation for the same graph would be {n} + {n/k}, a combination of the Schläfli symbols for the regular n-gon and star polygon from which the graph is formed. The Petersen graph itself is G(5, 2) or {5} + {5/2}. Any generalized Petersen graph can also be constructed from a voltage graph with two vertices, two self-loops, and one ...
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.
The two graphs and , are nonplanar, as may be shown either by a case analysis or an argument involving Euler's formula. Additionally, subdividing a graph cannot turn a nonplanar graph into a planar graph: if a subdivision of a graph has a planar drawing, the paths of the subdivision form curves that may be used to represent the edges of itself ...
The Petersen graph (on the left) and its complement graph (on the right).. In the mathematical field of graph theory, the complement or inverse of a graph G is a graph H on the same vertices such that two distinct vertices of H are adjacent if and only if they are not adjacent in G.
A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible. In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color.