Consider the following graph G with five vertices a, b, c, d, and e: a b d e Select all true statements about G. select one or more: O G is not connected. O There is a cycle of length 6 in G. O (c, d, b, e, a) is a Hamiltonian path in G.

1. How many Hamiltonian paths exist for the following graph? Recall that for an undirected graph the reverse path is the same as the original path; for example, E,C,D,A,F,B is the same thing as B,F,A,D,C,E. E A B

Which of the following sequences of letters represents a Hamiltonian circuit of the graph below? B F G A H D ADHGFEBCIA FGHDAJCBADEF CBEFGHDAJCBA ABCFEBADHGFCJ

Find a HAMILTONIAN CIRCUIT of the graph below (Give a sequence of letters to describe the path (e.g. A, D, E, B, . etc) B C OD E F

Which of the following graphs have hamiltonian circuits? I M N H K OA 0 OC G L I P F H G H E F D G F I C H OD E B I D

Determine two different Hamilton paths in the following graph. A J CD Select all that apply A. AEIJ, F, B, C, D, H, G, K, L B. B, F, E, I, J, K, L, H, G, B, D, A □ C. J, K, L, H, D, C, G, F, B, A, E, I D. E, F, G, H, L, K, J, I, E, A, B, C L

Find a Hamiltonian circuit in the following graph. (There is more than one possible answer.) Either clearly mark the path you take with directional arrows and numbers or list the vertices in the order that you visit them.

3. Given the graph below; a) Determine if the graph has an Eulerian Cycle without using trial-and-error to find it. Thoroughly justify your answer. b) Determine if the graph has an Eulerian Path without using trial-and-error to find it. Thoroughly justify your answer. c) Find a Hamiltonian Cycle starting at vertex A. Draw the Hamiltonian Cycle on the graph and list the vertices of the cycle. F M H Note: A Hamiltonian Cycle is a simple cycle that traverses all vertices. A simple cycle starts at a vertex, visits other vertices once then returns to the starting vertex. For instance, if the vertices of Ks were labeled A, B, C, D, E, one Hamiltonian Cycle would be ABCDEA.

Which of the following graphs have Euler circuits or Euler trails? E D R A: Has Euler trail. A: Has Euler circuit. G V C B I H C: Has Euler trail. C: Has Euler circuit. U V U B: Has Euler trail. B: Has Euler circuit. H V D: Has Euler trail. D: Has Euler circuit.

I have to prove the following Corollary: "Let u and v be vertices of a 2-connected graph G. Then there is a cycle of G that contains both u and v.”   My idea of the proof is the following: Given vertices u and v, I want to show that there is a cycle containing both. Then by contradiction, I could assume that there exists u and v sucht that there are no cycles C including u and v.  However, I'm not quite sure that this is the correct approach. I would appreciate some help to prove this corollary. Thank youuu