Math 2.0

Combinatorics Definition

Cycle

Builds on Circuit

Definition

Let GG be a graph. A cycle in GG is a circuit (w0,w1,…,wk)(w_0, w_1, \dots, w_k) whose vertices w1,…,wkw_1, \dots, w_k are pairwise distinct.

Remarks

The only repeated vertex is w0=wkw_0 = w_k. In a simple graph, each cycle has length k≥3k \ge 3. The vertices and edges of a cycle of length kk form a subgraph isomorphic to the cycle graph CkC_k. A graph with no cycle is acyclic.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Graph, definition, no evidence levelGraphdefinitionWalk, definition, no evidence levelWalkdefinitionTrail, definition, no evidence levelTraildefinitionCircuit, definition, no evidence levelCircuitdefinitionCycle, definition, no evidence levelCycledefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Graph definition
  2. Walk definition
  3. Trail definition
  4. Circuit definition
  5. Cycle definition

Where this leads

25 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cycle_(graph_theory), Circuit and cycle
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