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1. Show that the tree-order associated with a rooted tree T is indeed a partial order on V(T), and verify the claims made about this partial order in the text.
Hint: Theorem 1.5.1.
2. Do the partition classes of a regular bipartite graph always have the same size?
Hint: Count the edges.
3. Show that a graph is bipartite if and only if every induced cycle has even length.
Hint: Show that if a graph contains any odd cycle at all it also contains an induced one.
4. Find a function f : N → N such that, for all k ∈ N, every graph of average degree at least f(k) has a bipartite subgraph of minimum degree at least k.
Hint: Given a graph G, how would you split its vertex set into two parts A and B so that the bipartite graph H defined by the A-B edges of G has minimum degree as large as possible? To find f, apply this method to a suitable subgraph G of a given graph G’, and determine how large d(G’) must be to ensure that δ(H) ≥ k.
A volcano fills the volume between the graphs z=0 and z=1/(x^2+y^2)^23 , and outside the cylinder x^2+y^2=1. Find the volume of this volcano.
A totem pole has representations of n animals arranged in a vertical column
Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. The insurance company knows that, last year, the life expectancy of its
At East Zone University (EZU) there are 565 students taking College Algebra or Calculus. 327 are taking College Algebra, 266 are taking Calculus, and 28 are taking both College Algebra and Calculus. How many are taking Calculus but not Algebra?
a rectangular field is five times as long as it is wide. if the perimiter of the field is 480yards, what are the field's dimensions?
Probability of sports, Eighty seven cities were surveyed to determine sports teams. The survey found that
a baseball is hit 3 feet above home plate. It will reach its highest point of 83 feet 125 feet from home plate.
What is the probability that the tire you recently purchased will last at most 81,000 km.?
A rabbit and a turtle have a foot race. The Turtle receives a 100 foot head start. The rabbit is tens times as fast as the turtle. How far must the rabbit run to catch the turtle?
A computer processes jobs on a first-come first-serve basis in a time-sharing environment. The jobs have Poisson arrival rates with an average of six minutes between arrivals.
The distance between the building and the pole is 189 ft, and the pole casts a shadow that is 4.25 ft long. How tall is the building?
a moving truck rental company has a total of 24 trucks in its lot. The trucks are either 4-wheelers or 18-wheelers. How many of each kind of truck does the company have if the total number of wheels is 138?
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