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A graph that contains a Hamiltonian cycle is called a Hamiltonian graph. Similar notions may be defined for directed graphs , where each edge (arc) of a path or cycle can only be traced in a single direction (i.e., the vertices are connected with arrows and the edges traced "tail-to-head").
A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. [2]
Each n-dimensional De Bruijn graph is the line digraph of the (n – 1)-dimensional De Bruijn graph with the same set of symbols. [4] Each De Bruijn graph is Eulerian and Hamiltonian. The Euler cycles and Hamiltonian cycles of these graphs (equivalent to each other via the line graph construction) are De Bruijn sequences.
The Hirotaka graph, with 7 and size 12, is the Harris graph with the smallest order. [1] [2] The first appearance of this graph was in a paper as an example of a non-Hamiltonian, tough graph. [6] Douglas Shaw proved it to be minimal by showing all Eulerian graphs of order 6 or lower were not Hamiltonian and tough.
To improve the lower bound, a better way of creating an Eulerian graph is needed. By the triangle inequality, the best Eulerian graph must have the same cost as the best travelling salesman tour; hence, finding optimal Eulerian graphs is at least as hard as TSP. One way of doing this is by minimum weight matching using algorithms with a ...
In one direction, the Hamiltonian path problem for graph G can be related to the Hamiltonian cycle problem in a graph H obtained from G by adding a new universal vertex x, connecting x to all vertices of G. Thus, finding a Hamiltonian path cannot be significantly slower (in the worst case, as a function of the number of vertices) than finding a ...
[2] [3] He references Euler's Letters to a German Princess. [7] [a] In Hamilton's illustration of the four categorical propositions [8] which can occur in a syllogism as symbolized by the drawings A, E, I, and O are: A: The Universal Affirmative Example: "All metals are elements." E: The Universal Negative Example: "No metals are compound ...
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.