Item 1.

Which of the graphs below have Euler paths? Which have Euler circuits?

A graph with six vertices.  Four vertices form a square, with another vertex centered inside the square, and the last vertex centered above the square.  Edges connect the corners of the square, and connect each corner to the center vertex.  The top two vertices of the square is adjacent to the vertex above the square.
A graph with five vertices: four arranged in a square with the last one at the center of the square.  Edges connect neighboring vertices of the square, and the center vertex is adjacent to each vertex of the square.
A graph with seven vertices.  Four vertices form the corners of a square, the remaining three are in a middle row, with one to the left, one in the center, and one to the right of the square.  Edges form the sides of the square.  Each vertex in the square is adjacent to the two middle-row vertices closest to it.
Three vertices aligned in a vertical column left of a single vertex on the right.  Edges connect the vertex on the right to each vertex on the left.  Among the vertices on the left, two arced edges connect the bottom vertex to the center vertex, and two more connect the center vertex to the top vertex.
in-context