Search results
Results from the WOW.Com Content Network
This tour corresponds to a Hamiltonian cycle in the line graph L(G), so the line graph of every Eulerian graph is Hamiltonian. Line graphs may have other Hamiltonian cycles that do not correspond to Euler tours, and in particular the line graph L(G) of every Hamiltonian graph G is itself Hamiltonian, regardless of whether the graph G is ...
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]
In graph theory, a Harris graph is defined as an Eulerian, tough, non-Hamiltonian graph. [1] [2] Harris graphs were introduced in 2013 when, at the University of Michigan, Harris Spungen conjectured that any graph which is both tough and Eulerian is sufficiently Hamiltonian.
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.
[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 ...
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 ...
The Hamiltonian paths are in one-to-one correspondence with the minimal feedback arc sets of the tournament. [5] Rédei's theorem is the special case for complete graphs of the Gallai–Hasse–Roy–Vitaver theorem, relating the lengths of paths in orientations of graphs to the chromatic number of these graphs. [6]
A graph that can be proven non-Hamiltonian using Grinberg's theorem. In graph theory, Grinberg's theorem is a necessary condition for a planar graph to contain a Hamiltonian cycle, based on the lengths of its face cycles. If a graph does not meet this condition, it is not Hamiltonian.