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A connected graph has an Euler cycle if and only if every vertex has even degree. 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]
A graph that contains a Hamiltonian path is called a traceable graph. A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once.
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.
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 a tough, Eulerian graph would be sufficient to be Hamiltonian. [3]
An Eulerian path is a walk that uses every edge of a graph exactly once. An Eulerian circuit (also called an Eulerian cycle or an Euler tour) is a closed walk that uses every edge exactly once. An Eulerian graph is a graph that has an Eulerian circuit. For an undirected graph, this means that the graph is connected and every vertex has even degree.
The cycle space, also, has an algebraic structure, but a more restrictive one. The union or intersection of two Eulerian subgraphs may fail to be Eulerian. However, the symmetric difference of two Eulerian subgraphs (the graph consisting of the edges that belong to exactly one of the two given graphs) is again Eulerian. [1]
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 ...
Ore's theorem is a generalization of Dirac's theorem that, when each vertex has degree at least n/2, the graph is Hamiltonian. For, if a graph meets Dirac's condition, then clearly each pair of vertices has degrees adding to at least n. In turn Ore's theorem is generalized by the Bondy–Chvátal theorem.