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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.
Here are cycle graphs for those two groups, where we choose to generate the green cycle on the left and to generate that cycle on the right: In the right-hand graph, the green cycle, after moving from 1 to σ τ {\displaystyle \sigma \tau } , moves next to σ 6 , {\displaystyle \sigma ^{6},} because
A graph is bipartite if its vertices can be colored with two different colors such that each edge has one endpoint of each color. A graph is cubic (or 3-regular) if each vertex is the endpoint of exactly three edges. Finally, a graph is Hamiltonian if there exists a cycle that passes through each of its vertices exactly once. Barnette's ...
A directed cycle graph of length 8. A directed cycle graph is a directed version of a cycle graph, with all the edges being oriented in the same direction. In a directed graph, a set of edges which contains at least one edge (or arc) from each directed cycle is called a feedback arc set.
A graph has a k-coloring if and only if it has an acyclic orientation for which the longest path has length at most k; this is the Gallai–Hasse–Roy–Vitaver theorem (Nešetřil & Ossona de Mendez 2012). For planar graphs, vertex colorings are essentially dual to nowhere-zero flows. About infinite graphs, much less is known.
A cycle (black) with two chords (green). As for this part, the graph is chordal. However, removing one green edge would result in a non-chordal graph. Indeed, the other green edge with three black edges would form a cycle of length four with no chords.
In graph theory, a branch of mathematics, a cycle basis of an undirected graph is a set of simple cycles that forms a basis of the cycle space of the graph. That is, it is a minimal set of cycles that allows every even-degree subgraph to be expressed as a symmetric difference of basis cycles. A fundamental cycle basis may be formed from any ...
Pseudoforest, a directed or undirected graph in which every connected component includes at most one cycle; Cycle graph, a graph that has the structure of a single cycle; Pancyclic graph, a graph that has cycles of all possible lengths; Cycle detection (graph theory), the algorithmic problem of finding cycles in graphs; Other similarly-named ...