Which of the following graphs has an Eulerian circuit?

Answer: The complement of a cycle on 25 vertices has an Eulerian circuit.

Key criterion for an Eulerian circuit: a graph has an Eulerian circuit exactly when every vertex has even degree and the graph is connected (considering only the edges present).

• Complement of a 25-vertex cycle: Each vertex in the cycle has degree 2, so in the complement each vertex has degree 24 - 2 = 22, which is even. For n = 25 (n ≥ 6), the complement of the cycle is connected (any two vertices are either adjacent in the complement or have a common neighbor in the complement), so both conditions are met and an Eulerian circuit exists.

• Complete graph on 90 vertices: Each vertex has degree 90 - 1 = 89, which is odd. Because not all vertices have even degree, this graph does not have an Eulerian circuit.

• Any k-regular graph with even k: While every vertex has even degree, the graph might be disconnected. Eulerian circuit requires connectedness as well. For example, two disjoint cycles are 2-regular (k = 2, even) but there is no single Eulerian circuit that covers both components.

• None of the above: This is incorrect because the first listed graph (the complement of a 25-vertex cycle) does satisfy the Eulerian conditions.

Therefore the correct choice is the complement of a cycle on 25 vertices.