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Chapter 11: Problem 1
List three situations, different from those in this section, where a graph could prove useful.
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Let \(G\) be a loop-free undirected graph where \(\Delta=\max _{v e v}\\{\) deg \((v)\\}\). a) Prove that \(\chi(G) \leq \Delta+1\). b) Find two types of graphs \(G\) where \(\chi(G)=\Delta+1\).
a) How many paths of length 4 are there in the complete graph \(K_{7}\) ?
(Remember that a path such as \(v_{1} \rightarrow v_{2} \rightarrow v_{3}
\rightarrow v_{4} \rightarrow v_{5}\) is considered to be the same as the path
\(\left.v_{5} \rightarrow v_{4} \rightarrow v_{3} \rightarrow v_{2} \rightarrow
v_{1} .\right)\)
b) Let \(m, n \in \mathbf{Z}^{+}\)with \(m
For \(n \geq 3\), let \(C_{n}\) denote the undirected cycle on \(n\) vertices. The graph \(\overline{C_{n}}\), the complement of \(C_{n}\), is often called the cocycle on \(n\) vertices. Prove that for \(n \geq 5\) the cocycle \(\bar{C}_{n}\) has a Hamilton cycle.
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.)
a) Let \(G=(V, E)\) be a loop-free connected graph with \(|V| \geq 11\). Prove that either \(G\) or its complement \(\bar{G}\) must be nonplanar. b) The result in part (a) is actually true for \(|V| \geq 9\), but the proof for \(|V|=9,10\), is much harder. Find a counterexample to part (a) for \(|V|=8\).
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