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[1] [2] Douglas Shaw proved it to be minimal by showing all Eulerian graphs of order 6 or lower were not Hamiltonian and tough. Java code written by Shubhra Mishra and Marco Troper proved it unique. [2] The first Harris graph discovered was the Shaw graph, which has order 9 and size 14.
An Eulerian trail, [note 1] or Euler walk, in an undirected graph is a walk that uses each edge exactly once. If such a walk exists, the graph is called traversable or semi-eulerian. [3] An Eulerian cycle, [note 1] also called an Eulerian circuit or Euler tour, in an undirected graph is a cycle that uses each edge exactly once
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.
Euler diagram illustrating that the set of "animals with four legs" is a subset of "animals", but the set of "minerals" is a disjoint set (it has no members in common) with "animals" Euler diagram showing the relationships between different Solar System objects
All Eulerian circuits are also Eulerian paths, but not all Eulerian paths are Eulerian circuits. Euler's work was presented to the St. Petersburg Academy on 26 August 1735, and published as Solutio problematis ad geometriam situs pertinentis (The solution of a problem relating to the geometry of position) in the journal Commentarii academiae ...
Action principles are "integral" approaches rather than the "differential" approach of Newtonian mechanics.[2]: 162 The core ideas are based on energy, paths, an energy function called the Lagrangian along paths, and selection of a path according to the "action", a continuous sum or integral of the Lagrangian along the path.
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.
A connected planar graph is Eulerian (has even degree at every vertex) if and only if its dual graph is bipartite. [32] A Hamiltonian cycle in a planar graph G corresponds to a partition of the vertices of the dual graph into two subsets (the interior and exterior of the cycle) whose induced subgraphs are both trees.