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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 \(C_4.\)

    graph_bowtie.svg
    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.

    This page titled 5.5: Cycles was last modified on Tue, 01 Sep 2026 03:34:21 GMT and is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Mark A. Fitch via source content that was edited to the style and standards of the LibreTexts platform.