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A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black. In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges).
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding ...
Snark (graph theory) Sparse graph. Sparse graph code; Split graph; String graph; Strongly regular graph; Threshold graph; Total graph; Tree (graph theory). Trellis (graph) Turán graph; Ultrahomogeneous graph; Vertex-transitive graph; Visibility graph. Museum guard problem; Wheel graph
The circuit rank controls the number of ears in an ear decomposition of a graph, a partition of the edges of the graph into paths and cycles that is useful in many graph algorithms. In particular, a graph is 2-vertex-connected if and only if it has an open ear decomposition. This is a sequence of subgraphs, where the first subgraph is a simple ...
A graph with edges colored to illustrate a closed walk, H–A–B–A–H, in green; a circuit which is a closed walk in which all edges are distinct, B–D–E–F–D–C–B, in blue; and a cycle which is a closed walk in which all vertices are distinct, H–D–G–H, in red.
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]
The Hamiltonian path problem is a topic discussed in the fields of complexity theory and graph theory. It decides if a directed or undirected graph , G , contains a Hamiltonian path , a path that visits every vertex in the graph exactly once.
In graph theory, a walk-regular graph is a simple graph where the number of closed walks of any length from a vertex to itself does only depend on but not depend on the choice of vertex. Walk-regular graphs can be thought of as a spectral graph theory analogue of vertex-transitive graphs .