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Chapter 11: Problem 5
Let \(G=(V, E)\) be an undirected graph, where \(|V| \geq 2\). If every induced subgraph of \(G\) is connected, can we identify the graph \(G ?\)
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List three situations, different from those in this section, where a graph could prove useful.
Let \(G=(V, E), H=\left(V^{\prime}, E^{\prime}\right)\) be undirected graphs with \(f: V \rightarrow V^{\prime}\) establishing an isomorphism between the graphs. a) Prove that \(f^{-1}: V^{\prime} \rightarrow V\) is also an isomorphism for \(G\) and \(H\). b) If \(a \in V\), prove that \(\operatorname{deg}(a)\) (in \(G\) ) \(=\operatorname{deg}(f(a))\) (in \(H\) ).
a) At the J. \& J. Chemical Company, Jeannette has received three shipments that contain a total of seven different chemicals. Furthermore, the nature of these chemicals is such that for all \(1 \leq i \leq 5\), chemical \(i\) cannot be stored in the same storage compartment as chemical \(i+1\) or chemical \(i+2 .\) Determine the smallest number of separate storage compartments that Jeannette will need to safely store these seven chemicals. b) Suppose that in addition to the conditions in part (a), the following four pairs of these same seven chemicals also require separate storage compartments: 1 and 4,2 and 5,2 and 6, and 3 and 6 . What is the smallest number of storage compartments that Jeannette now needs to safely store the seven chemicals?
Prove that for any \(n \in \mathbf{Z}^{+}\)there exists a loop-free connected undirected graph \(G=\) \((V, E)\) where \(|V|=2 n\) and which has two vertices of degree \(i\) for every \(1 \leq i \leq n\).
For \(n \geq 3\), recall that the wheel graph, \(W_{n}\), is obtained from a cycle of length \(n\) by placing a new vertex within the cycle and adding edges (spokes) from this new vertex to each vertex of the cycle. a) What relationship is there between \(\chi\left(C_{n}\right)\) and \(\chi\left(W_{n}\right)\) ?b) Use part (e) of Exercise 13 to show that $$ P\left(W_{n}, \lambda\right)=\lambda(\lambda-2)^{n}+(-1)^{n} \lambda(\lambda-2) \text {. } $$ c) i) If we have \(k\) different colors available, in how many ways can we paint the walls and ceiling of a pentagonal room if adjacent walls, and any wall and the ceiling, are to be painted with different colors? ii) What is the smallest value of \(k\) for which such a coloring is possible? iii) Answer parts (i) and (ii) for a hexagonal room. (The reader may wish to compare part (c) of this exercise with Exercise 6 in the Supplementary Exercises of Chapter 8.)
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