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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 ]
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 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 ...
Herz (1968) defines the cyclability of a graph as the largest number k such that every k vertices belong to a cycle; the hypohamiltonian graphs are exactly the graphs that have cyclability n − 1. Similarly, Park, Lim & Kim (2007) define a graph to be ƒ-fault hamiltonian if the removal of at most ƒ vertices leaves a Hamiltonian subgraph.
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.
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 ...
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 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.