�✁ ❪ ❭ ◆ ❙ ▼ ❘ ❪ ❭ ◆ ❙ ❫ ❭ ❭ ❘ ❳ ❖ ❩ ❲ P ❯ ❨ ❘ ❲ P ❯ ❫ ✰✱✲ ❚ ❃ ✼ ✱ ❃ ❇ ❆ ✳ ❅ ❀ ❁❄ ✷ ✵ ❂ ✳ ❀❁ ✿ ✾ ✽ ✶ ✻✼ ✺ ✹ ✸ ✶✷ ✵ ✴ ◆ ◗ ❃ ✱ ✂✄ ☎ �✁ ✼ ✹ ❈ ✺ ✱ ❃ ✹ ✼ ❃ ✝ ❇ ❆ ✳ ❅ ❀ ❁❄ ✷ ✵ ❃ ❂ ❀❁ ✄ ✆ ❙ ◗ ◗ ❖ ❘ ◗ P ❖ ◆ ▼ ▲ ❑ ❖ ❖ ❋ ❘ ◗ P ❖ ◆ ▼ ▲ ❑ ✂ ❍ ☎ ● ✹ ✱ ✾ ▼ ✵ ✴ ✳ ✰✱✲ ❑ ❲ ◆ ✐ ❤ ❣ ❢ ❭ ✸ ❡ ❭ ❘ ◗ P ❖ ◆ ▼ ▲ ❑ ❲ ✶✷ ✹ ❑ ✳ ✹ ❈ ✺ ✱ ❃ ✹ ✼ ✱ ❃ ❇ ❆ ❅ ✺ ❀ ❁❄ ✷ ✵ ❃ ❂ ❀❁ ✿ ✾ ✽ ✶ ✻✼ ❲ ❳ ✺ ❋ ❜ ❛ ❊ ✁ ❵ ☎ ✆ ❴ ❍ ☎ ● ✆ ❝❞ ✝ ✄ ✆ ☎ ✁ ❋ ❊ ❉ ✻ ✹ ❈ ❛ ❑ ❲ ❳ P ❯ ❚ ❲ ❯ ❲ ❯ ❭ ❲ ❯ ❭ ❲ ❯ P ▼ ❭ ▲ ❳ ❲ P ❯ ❚ ▲ ▲ ❲ ✿ ✆ ✽ ❃ ✆ ☎ ✁ ❋ ❊ ❉ ❈ ✹ ❈ ✺ ✱ ✹ ✝ ✼ ✱ ❃ ❇ ❆ ✳ ❅ ❀ ❁❄ ✷ ✵ ❃ ✄ ✆ ❀❁ ❖ ❖ ❘ ◗ P ❖ ◆ ▼ ▲ ❑ ❖ ◗ ✶ ❋ ❘ ◗ P ❖ ◆ ▼ ▲ ❑ ✂ ❍ ☎ ● ❂ ✿ ❙ ✠ ✘ ✗ ✖ ✔✕ ✓ ✒ ✑ ✏ ✎ ☛ ✡ ✂ ✕ ✆ ☎ ✟ ✝ ✄ ✟ ☎ ✝ ✞ ✆ ✆✝ ☎ ✙ ✚ ✾ ✭ ✽ ✶ ✻✼ ✺ ✹ ✸ ✶✷ ✵ ✴ ✳ ✰✱✲ ✮✯ ✪✬ ✔ ✩✪✫ ★ ✧ ✤ ✗ ✦ ✤✥ ✗ ✣ ✢ ✜ ✛ ◗ ✺ ❙ ❘ ▼ ▼ ▲ ❚ ❯ ❑ P ❯ ❫ ❘ ❭ ◆ ❳ ❙ ▼ ▼ ❳ ❘ ◆ ❭ ◗ ❳❨ ❖ ◗ ❚ ❚ ❯ P ❘ ❲ ◗ ❖ ❯ P ❲ ❩ P ❖ ❳ ❙ ◆ ❯ ❩ ❖ P ✹ ❪ P ❖ ❩ ❲ ❯ ▼ ❳❨ ❲ P ✺ ❚ ✻✼ ◗ ❬ ❯ ❲ ❳ ❭ ✰✱✲ ◆ ❙ ✳ ✴ ❫ ❭ ❪ ✸ ✵ ❭ ✶✷ ■✍❏ ☞✍✌ A simple graph consists of , a nonempty set of vertices, and , a set of unordered pairs of distinct elements of called edges. A multigraph consists of a set of vertices, a set of edges, and a function from to . The edges ❚❱❯ and are called multiple or parallel edges if . A pseudograph consists of a set of vertices, a set of edges, and a function from to . An edge is a loop if for some . ❚❱❯ ■✍❏ ■✍❏ Two vertices and in an undirected graph are called adjacent (or neighbors ) in if is an edge of . If , the edge is A directed graph consists of a set of vertices and a set of ❚❱❯ called incident with the vertices and . The edge is also said to edges that are ordered pairs of elements of . connect and . The vertices and are called endpoints of the edges . A directed multigraph consists of a set of vertices, a set The degree of a vertex in an undirected graph is the number of edges incident with it, except that a loop at a vertex contributes twice to the of edges, and a function from to . The edges and are multiple edges if . degree of that vertex. The degree of the vertex is denoted by deg( ). The Handshaking Theorem : Let be an undirected graph with edges. Then deg ❘❦❥ Theorem : An undirected graph has an even number of vertices of odd degree.
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❷ ❅ ✶ ✽ ✾ ✿ ❀❁ ❂ ❃ ✵ ✷ ❁❄ ❀ ✳ ✺ ❆ ❇ ❃ ✱ ✼ ✹ ❃ ✱ ✺ ❈ ✹ ✻✼ ✹ ✟ ❑ ❜ ✆ ☎ ③ ✁ ☎ ✄ ✟ ✁ ✝ ✂ ❸ ✸ ▲ ▲ ❑ ✰✱✲ ✳ ✴ ✵ ✶✷ An Euler circuit in a graph is a simple circuit containing every edge of . Theorem: A connected multigraph has an Euler circuit if and only if each of its vertices has even degree.
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