Definition 5.5.1. 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 \(v_0=v_k.\)
Section 5.5 Cycles
Subsection 5.5.1 Terminology
Definition 5.5.2. 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 5.5.3. Eulerian Circuit.
A circuit is Eulerian if and only if it uses every edge in the graph and no edge is used twice.
Definition 5.5.4. Hamiltonian Circuit.
A circuit is Hamiltonian if and only if it uses every vertex in that graph and no edge is used twice.
Subsection 5.5.2 Practice
Checkpoint 5.5.5.
Draw a cycle of length 4. Note this is denoted \(C_4.\)
Checkpoint 5.5.7.
Find a circuit that is not a cycle in Figure 5.5.6.
Checkpoint 5.5.8.
Determine which graphs in Figure Figure 5.2.43 have Eulerian circuits.
Checkpoint 5.5.9.
Determine which graphs in Figure Figure 5.2.43 have Hamiltonian circuits.
