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The Petersen graph is hypo-Hamiltonian: by deleting any vertex, such as the center vertex in the drawing, the remaining graph is Hamiltonian. This drawing with order-3 symmetry is the one given by Kempe (1886). The Petersen graph has a Hamiltonian path but no Hamiltonian cycle. It is the smallest bridgeless cubic graph with no Hamiltonian cycle.
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 complete list of hypohamiltonian graphs with at most 17 vertices is known: [11] they are the 10-vertex Petersen graph, a 13-vertex graph and a 15-vertex graph found by computer searches of Herz (1968), and four 16-vertex graphs. There exist at least thirteen 18-vertex hypohamiltonian graphs.
All Hamiltonian graphs are biconnected, but a biconnected graph need not be Hamiltonian (see, for example, the Petersen graph). [9] An Eulerian graph G (a connected graph in which every vertex has even degree) necessarily has an Euler tour, a closed walk passing through each edge of G exactly once.
The Petersen graph is an example: it is the smallest 3-regular graph with no cycles of length shorter than 5. Chapter seven is on hypohamiltonian graphs, the graphs that do not have a Hamiltonian cycle through all vertices
If a cubic graph is chosen uniformly at random among all n-vertex cubic graphs, then it is very likely to be Hamiltonian: the proportion of the n-vertex cubic graphs that are Hamiltonian tends to one in the limit as n goes to infinity. [10] David Eppstein conjectured that every n-vertex cubic graph has at most 2 n/3 (approximately 1.260 n ...
It can also be applied to analyze the Hamiltonian cycles of certain non-planar graphs, such as generalized Petersen graphs, by finding large planar subgraphs of these graphs, using Grinberg's theorem to show that these subgraphs are non-Hamiltonian, and concluding that any Hamiltonian cycle must include some of the remaining edges that are not ...
It is well-known that the Petersen graph is not Hamiltonian, but it was long conjectured that this was the sole exception and that every other connected Kneser graph K(n, k) is Hamiltonian. In 2003, Chen showed that the Kneser graph K ( n , k ) contains a Hamiltonian cycle if [ 7 ]