answer:A Hamiltonian cycle is a cycle that visits each vertex exactly once and returns to the starting vertex. In a complete graph K7, there are 7 vertices, and each vertex is connected to every other vertex. One possible Hamiltonian cycle in K7 is as follows: 1 → 2 → 3 → 4 → 5 → 6 → 7 → 1 This cycle visits each vertex exactly once and returns to the starting vertex (1). Since K7 is a complete graph, there are multiple Hamiltonian cycles possible, but this is one valid example.

answer:To determine whether the graph G contains a Hamiltonian cycle, we need more information about the specific connections between the vertices. A Hamiltonian cycle is a cycle that visits each vertex exactly once and returns to the starting vertex. However, we can analyze the general properties of a graph with 6 vertices and 8 edges to see if it is likely to contain a Hamiltonian cycle. A graph with 6 vertices can have a maximum of 15 edges (when it is a complete graph). In this case, we have 8 edges, which is slightly more than half of the maximum possible edges. This means that the graph is not very sparse, increasing the likelihood of a Hamiltonian cycle. Additionally, we can use Dirac's theorem as a sufficient condition for the existence of a Hamiltonian cycle. Dirac's theorem states that if a graph G has n vertices (n ≥ 3) and the degree of each vertex is at least n/2, then G has a Hamiltonian cycle. In our case, n = 6, so n/2 = 3. If each vertex has a degree of at least 3, then according to Dirac's theorem, the graph G would have a Hamiltonian cycle. However, without more information about the specific connections between vertices, we cannot definitively determine whether G contains a Hamiltonian cycle or not. If you provide the specific connections between the vertices, we can determine the existence of a Hamiltonian cycle and show it on the graph.

answer:To determine if a Hamiltonian cycle exists in a graph G with 7 vertices and 9 edges, we need to consider the degree of each vertex. A Hamiltonian cycle is a cycle that visits each vertex exactly once and returns to the starting vertex. A necessary condition for a graph to have a Hamiltonian cycle is that each vertex must have a degree of at least 2. This is because, in a Hamiltonian cycle, each vertex must be connected to two other vertices (one before and one after it in the cycle). However, this condition is not sufficient to guarantee the existence of a Hamiltonian cycle. For example, consider a graph with 7 vertices (A, B, C, D, E, F, G) and the following edges: 1. A-B 2. B-C 3. C-D 4. D-E 5. E-F 6. F-G 7. G-A 8. A-C 9. E-G This graph has 9 edges, and each vertex has a degree of at least 2. However, there is no Hamiltonian cycle in this graph. To see this, note that if we start at vertex A, we can go to B, C, D, E, and F, but then we cannot go to G without retracing an edge (F-G or E-G), and we cannot return to A without skipping G. So, in this particular graph with 7 vertices and 9 edges, there is no Hamiltonian cycle. Note that this does not mean that no graph with 7 vertices and 9 edges can have a Hamiltonian cycle, but it shows that it is not guaranteed.

answer:A Hamiltonian cycle is a cycle that visits each vertex exactly once and returns to the starting vertex. Let's try to find a Hamiltonian cycle in the given graph. We can start at vertex 1 and follow the edges in this order: (1,2), (2,3), (3,4), (4,5), (5,6), (6,1). This cycle visits each vertex exactly once and returns to the starting vertex 1. So, the Hamiltonian cycle in the graph is: 1-2-3-4-5-6-1.