5.5: Cycles

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Terminology

Definition \(\PageIndex{1}\): Circuit.

A graph is a circuit, or closed walk, if and only if the vertices can be labeled \(v_0, v_1, \ldots, v_k\) such that \(\{v_j,v_{j+1}\}\) is an edge for all \(j\) and \(v_0=v_k.\)

Definition \(\PageIndex{2}\): Cycle.

A graph is a cycle if and only if it is a circuit such that no vertex is used twice.

A cycle on \(n\) vertices is denoted \(C_n.\) Like walk and path terminology for cycles varies widely.

Definition \(\PageIndex{3}\): Eulerian Circuit.

A circuit is Eulerian if and only if it uses every edge in the graph and no edge is used twice.

Definition \(\PageIndex{4}\): Hamiltonian Circuit.

A circuit is Hamiltonian if and only if it uses every vertex in that graph and no edge is used twice.

Practice

Checkpoint \(\PageIndex{5}\)

Draw a cycle of length 4. Note this is denoted \(C_4.\)

Figure \(\PageIndex{6}\) Graph Containing Circuits and Cycles

Checkpoint \(\PageIndex{5}\)

Find a circuit that is not a cycle in Figure \(\PageIndex{6}\)

Checkpoint \(\PageIndex{7}\)

Determine which graphs in Figure 5.2.43 have Eulerian circuits.

Checkpoint \(\PageIndex{9}\)

Determine which graphs in Figure 5.2.43 have Hamiltonian circuits.