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Chapter 10: Problem 50
How many subgraphs with at least one vertex does \(K_{2}\) have?
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Suppose that a connected planar graph has eight vertices, each of degree three. Into how many regions is the plane divided by a planar representation of this graph?
In an old puzzle attributed to Alcuin of York \((735-804),\) a farmer needs to carry a wolf, a goat, and a cabbage across a river. The farmer only has a small boat, which can carry the farmer and only one object (an animal or a vegetable). He can cross the river repeatedly. However, if the farmer is on the other shore, the wolf will eat the goat, and, similarly, the goat will eat the cabbage. We can describe each state by listing what is on each shore. For example, we can use the pair \((F G, W C)\) for the state where the farmer and goat are on the first shore and the wolf and cabbage are on the other shore. [The symbol \(\emptyset\) is used when nothing is on a shore, so that \((F W G C, \emptyset)\) is the initial state. \(]\) a) Find all allowable states of the puzzle, where neither the wolf and the goat nor the goat and the cabbage are left on the same shore without the farmer. b) Construct a graph such that each vertex of this graph represents an allowable state and the vertices representing two allowable states are connected by an edge if it is possible to move from one state to the other using one trip of the boat. c) Explain why finding a path from the vertex representing \((F W G C, \emptyset)\) to the vertex representing \((\emptyset, F W G C)\) solves the puzzle. d) Find two different solutions of the puzzle, each using seven crossings. e) Suppose that the farmer must pay a toll of one dollar whenever he crosses the river with an animal. Which solution of the puzzle should the farmer use to pay the least total toll?
A knight is a chess piece that can move either two spaces horizontally and one space vertically or one space horizontally and two spaces vertically. That is, a knight on square \((x, y)\) can move to any of the eight squares \((x \pm 2, y \pm 1),(x \pm 1, y \pm 2),\) if these squares are on the chessboard, as illustrated here. A knight's tour is a sequence of legal moves by a knight starting at some square and visiting each square exactly once. A knight's tour is called reentrant if there is a legal move that takes the knight from the last square of the tour back to where the tour began. We can model knight's tours using the graph that has a vertex for each square on the board, with an edge connecting two vertices if a knight can legally move between the squares represented by these vertices. Show that the graph representing the legal moves of a knight on an \(m \times n\) chessboard, whenever \(m\) and \(n\) are positive integers, is bipartite.
Give a big-O estimate of the number of operations (comparisons and additions) used by Floyd鈥檚 algorithm to determine the shortest distance between every pair of vertices in a weighted simple graph with n vertices.
How many different channels are needed for six stations located at the distances shown in the table, if two stations cannot use the same channel when they are within 150 miles of each other? $$\begin{array}{|c|c|c|c|c|c|}\hline & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline 1 & {-} & {85} & {175} & {200} & {50} & {100} \\ \hline 2 & {85} & {-} & {125} & {175} & {100} & {160} \\ \hline 4 & {200} & {175} & {100} & {-} & {210} & {220} \\ \hline 5 & {50} & {100} & {200} & {210} & {-} & {100} \\\ \hline 6 & {100} & {160} & {250} & {220} & {100} & {-} \\ \hline\end{array}$$
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