Three couples arrive at the bank of a river. Each of the wives is jealous and does not trust her husband when he is with one of the other wives (and perhaps with other people), but not with her. How can six people cross to the other side of the river using a boat that can hold no more than two people so that no husband is alone with a woman other than his wife? Use a graph theory model.

Show that {(x, y) | x − y ∈ Q} is an equivalence relation on the set of real numbers, where Q denotes the set of rational numbers. What are [1], [1/2], and [π]?

Let A be the adjacency matrix of a graph G with n vertices. Let $Y=A+A^{2}+\cdots+A^{n-1}$ If some off-diagonal entry in the matrix Y is zero, what can you say about the graph G?

Determine whether each of these sequences is graphic. For those that are, draw a graph having the given degree sequence. a) 3, 3, 3, 3, 2 b) 5, 4, 3, 2, 1 c) 4, 4, 3, 2, 1 d) 4, 4, 3, 3, 3 e) 3, 2, 2, 1, 0 f) 1, 1, 1, 1, 1

a) If a tree has four vertices of degree 2, one vertex of degree 3, two of degree 4, and one of degree 5, how many pendant vertices does it have? b) If a tree T = (V, E) has v2 vertices of degree 2, v3 vertices of degree 3, …, and vm vertices of degree m, what are |V| and |E|?

Find an ordering of the tasks of a software project if the Hasse diagram for the tasks of the project is as shown in the given figure.

Show that a bipartite graph with an odd number of vertices does not have a Hamilton circuit.

Show that a tree has either one center or two centers that are adjacent.

Draw the game tree for the game of tic-tac-toe for the levels corresponding to the first two moves. Assign the value of the evaluation function mentioned in the text that assigns to a position the number of files containing no Os minus the number of files containing no Xs as the value of each vertex at this level and compute the value of the tree for vertices as if the evaluation function gave the correct values for these vertices.

For which graphs do depth-first search and breadth-first search produce identical spanning trees no matter which vertex is selected as the root of the tree? Justify your answer.

When must an edge of a connected simple graph be in every spanning tree for this graph?

Show that a center should be chosen as the root to produce a rooted tree of minimal height from an unrooted tree.

Suppose that e is an edge in a weighted graph that is incident to a vertex v such that the weight of e does not exceed the weight of any other edge incident to v. Show that there exists a minimum spanning tree containing this edge.

How many children does the root of the game tree for nim have and how many grandchildren does it have if the starting position is a) piles with four and five stones, respectively. b) piles with two, three, and four stones, respectively. c) piles with one, two, three, and four stones, respectively. d) piles with two, two, three, three, and five stones, respectively.