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Answer the questions below. Assume |V| is finite and positive. (a) [3 marks] Prove that for all undirected graphs G = (V, E), if C is a cycle in G and e is an edge in C, then removing e leaves C connected. Notice that this is used in Example 6.8, so you cannot use Example 6.8 as proof, nor can you use the fact that this is asserted without proof in the paragraph just before Example 6.8. (b) [3 marks] Prove or disprove: In every undirected graph G = (V, E) with all vertices having degree at least ?|V|/3?, for every 3 distinct vertices u, v, w ? V there is a path of length no more than 2 from u to v, or from v to w, or from w to u. (c) [3 marks] Prove or disprove: If graph G = (V, E) has an odd number of vertices with even degree, then |V| is odd. (d) [3 marks] Prove or disprove: Every undirected graph G = (V, E) with at least 13 vertices, all vertices having degree at least |V| - 7, is connected. (e) [3 marks] Prove that every undirected graph G = (V, E) with every vertex having degree at least 5, has a cycle. (f) [3 marks] Prove or disprove: If G = (V, E) is an undirected graph where every vertex has degree at least 4 and u ? V, then there are at least 64 distinct paths in G that start at u.

          Answer the questions below. Assume |V| is finite and positive.

(a) [3 marks] Prove that for all undirected graphs G = (V, E), if C is a cycle in G and e is an edge in C, then removing e leaves C connected. Notice that this is used in Example 6.8, so you cannot use Example 6.8 as proof, nor can you use the fact that this is asserted without proof in the paragraph just before Example 6.8.
(b) [3 marks] Prove or disprove: In every undirected graph G = (V, E) with all vertices having degree at least ?|V|/3?, for every 3 distinct vertices u, v, w ? V there is a path of length no more than 2 from u to v, or from v to w, or from w to u.
(c) [3 marks] Prove or disprove: If graph G = (V, E) has an odd number of vertices with even degree, then |V| is odd.
(d) [3 marks] Prove or disprove: Every undirected graph G = (V, E) with at least 13 vertices, all vertices having degree at least |V| - 7, is connected.
(e) [3 marks] Prove that every undirected graph G = (V, E) with every vertex having degree at least 5, has a cycle.
(f) [3 marks] Prove or disprove: If G = (V, E) is an undirected graph where every vertex has degree at least 4 and u ? V, then there are at least 64 distinct paths in G that start at u.

Answer the questions below. Assume |V| is finite and positive.

(a) [3 marks] Prove that for all undirected graphs G = (V, E), if C is a cycle in G and e is an edge in C, then removing e leaves C connected. Notice that this is used in Example 6.8, so you cannot use Example 6.8 as proof, nor can you use the fact that this is asserted without proof in the paragraph just before Example 6.8.
(b) [3 marks] Prove or disprove: In every undirected graph G = (V, E) with all vertices having degree at least ?|V|/3?, for every 3 distinct vertices u, v, w ? V there is a path of length no more than 2 from u to v, or from v to w, or from w to u.
(c) [3 marks] Prove or disprove: If graph G = (V, E) has an odd number of vertices with even degree, then |V| is odd.
(d) [3 marks] Prove or disprove: Every undirected graph G = (V, E) with at least 13 vertices, all vertices having degree at least |V| - 7, is connected.
(e) [3 marks] Prove that every undirected graph G = (V, E) with every vertex having degree at least 5, has a cycle.
(f) [3 marks] Prove or disprove: If G = (V, E) is an undirected graph where every vertex has degree at least 4 and u ? V, then there are at least 64 distinct paths in G that start at u.

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