# Definition: Closed Walks, Closed Trails, and Cycles

Let $$G(V,E,\gamma)$$ be a simple graph. * A walk (or trail) $$W^k=v_0e_0v_1e_1v_2e_2\ldots e_{k-1}x_{k}$$ in $$G$$ is called a closed walk (or a closed trail) if $$x_k=x_0$$. * A path $$C^k:=x_0x_1\ldots x_{k-1}x_k$$ with $$x_0=x_k$$, but $$x_0\neq x_i$$ for $$i=1,\ldots,k-1$$ is called a cycle. Note that from this definition it allows any cycle in a simple graph contains at least three different vertices. * An edge which joins two vertices of a cycle but is not itself an edge of the cycle is a chord of that cycle.

#### Examples:

In the figure above:

• $$abcgfcdefcdefgha$$ is a closed walk, but not a closed trail because the e.g. the edges $$gf$$, $$cd$$, etc. are passed more then once.
• $$abcdefcgha$$ is a closed trail, but not a cycle because the vertex $$c$$ is passed more then once.
• $$abcdefgha$$ is a cycle (every edge and every vertex is passed exactly once); the edges $$cg$$ and $$cf$$ are the chords of that cycle.
• $$abcgha$$ is a cycle without any chords.

Definitions: 1 2 3 4 5 6 7 8 9 10
Explanations: 11
Lemmas: 12
Proofs: 13 14 15 16 17 18 19 20 21
Propositions: 22 23 24
Theorems: 25 26 27

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### References

#### Bibliography

1. Aldous Joan M., Wilson Robin J.: "Graphs and Applications - An Introductory Approach", Springer, 2000
2. Diestel, Reinhard: "Graph Theory, 3rd Edition", Springer, 2005