Find three different spanning trees for the given graph. (There are many different correct answers).

The material applied between tiles is called grout. For the floor tiling, in which could the grout be applied be inning and ending at the same place, without going over any joint twice, and without lifting the tool? Justify each answer.

Determine whether the given graph is connected or disconnected. Then determine how many components the graph has.

The material applied between tiles is called grout. For the given floor tiling, in which could the grout be applied be inning and ending at the same place, without going over any joint twice, and without lifting the tool? Justify each answer.

The floor plan of a building is shown. Is it possible to start outside, walk through each door exactly once, and end up back outside? Justify your answer.

The vertices of a graph represent the countries on the continent (Western Europe), with edges representing border crossings between the countries. A traveler wants to travel over each border crossing exactly once, returning to the first country visited for his flight home to the United States. Determine whether an Euler circuit, a Hamilton circuit, or neither would solve the problem.