Euler Cycle

Description:

• A cycle that uses every edges in a graph exactly once
• A undirected graph 𝐺 = (𝑉, 𝐸) has an Euler cycle if and only if 𝐺 is connected and every 𝑣 ∈ 𝑉 has even degree.theorem
• A directed graph 𝐺 = (𝑉, 𝐸) has an Euler cycle if and only if 𝐺 is strongly connected and every 𝑣 ∈ 𝑉 has equal in-degree and out-degree.theorem
• Constructing Euler circuit:
  • Start from any vertex and find a sub-cycle, start and end at this vertex (it exists due to the even degrees assumption)
  • Remove all edge of this sub-cycle.
  • Repeat the process at an appropriate vertex until all edges were removed.
  • Combine the sub-cycles to form the Euler cycle.