Chapter 10: Q37SE (page 738)
Determine whether the given simple graph is orientable.
Yes, the graph is orientable due to the clockwise direction of the paths.
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Find the chromatic number of the given graph.
How many edges does a \({\rm{50}}\)-regular graph with \({\rm{100}}\)vertices have?
Given two chickens in a flock, one of them is dominant. This defines the pecking order of the flock. How can a tournament be used to model pecking order?
Devise an algorithm for finding the second shortest path between two vertices in a simple connected weighted graph.
Suppose that to generate a random simple graph with n vertices we first choose a real number p with\({\bf{0}} \le {\bf{p}} \le {\bf{1}}\). For each of the\({\bf{C}}\left( {{\bf{n,2}}} \right)\)pairs of distinct vertices we generate a random number x between 0 and 1.
If\({\bf{0}} \le {\bf{x}} \le {\bf{p}}\), we connect these two vertices with an edge; otherwise these vertices are not connected.```
a) What is the probability that a graph with m edges where\({\bf{0}} \le {\bf{m}} \le {\bf{C}}\left( {{\bf{n,2}}} \right)\)is generated?
b) What is the expected number of edges in a randomly generated graph with n vertices if each edge is included with probability p?
c) Show that if\({\bf{p = }}\frac{{\bf{1}}}{{\bf{2}}}\)then every simple graph with n vertices is equally likely to be generated.
A property retained whenever additional edges are added to a simple graph (without adding vertices) is called monotone increasing, and a property that is retained whenever edges are removed from a simple graph (without removing vertices) is called monotone decreasing.
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