Due May 18
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Practice problems:
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| Hand in:
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Due April 19
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Practice problems:
- Section 4.2, #1,2,3,10
- Section 5.1, #3,5,12
- Section 5.2, #1,2,3,4,5,12,
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Hand in:
- Section 4.2, #11, 16, 17
- Section 5.1, #8, 11
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Due April 1
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Practice problems:
- Section 4.1, 4,13, 20, 21, 22, 24
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Hand in:
- Section 4.1, #8, 11, 12, 25
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Due March 25
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Practice problems:
- Section 3.2, # 8, 9,11
- Section 3.3, #2, 3,5, 11
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Hand in:
- Section 3.2, #5
- Section 3.3, # 4, 9, 10
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Due March 22
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Practice problems:
- Section 3.1, #6, 13, 17, 18
- Draw all trees with 5 or fewer vertices.
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Hand in:
- Section 3.1, #14
- Suppose d1 >= d2 >= ...>=dn is the degree sequence
for a tree. Prove that when the tree has at leat 6 vertices, then
di <= the ceiling of (n-1)/i. Is this true for trees with
fewer
than 6 vertices?
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Due March 16
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Practice problems:
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Hand in:
- Section 3.1, #11, 12, 16
- Prove that if a graph has fewer edges than vertices then
at
least one of its connected components must be a tree.
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Due March 4
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Practice problems:
- Section 2.4, #8,10,14, 18
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Hand in:
- Section 2.4, #13
- Prove that if a graph is disconnected then its complement
must be connected.
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Due March 2
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Practice problems:
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Hand in:
- Section 2.3, #17
- Suppose that f: X->Y and g:Y->Z are bijective
functions. Prove that the composition of g and f is bijective.
- Prove that a 5 cycle can't be bipartite by doing a proof by
contradiction
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Due Feb. 26
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Practice problems:
- Section 1.1, #35 (a is not true), 38
- A bridge in a
graph is an edge in the graph whose removal disconnects the graph. Draw
a connected graph with a bridge. Are the vertices of the bridges even
or
odd?
- Does every disconnected graph have an isolated vertex?
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Hand in:
- Section 1.1,#37, 39
- A sequence of numbers is defined by f(0) = f(1) = 1 and for
n>1, f(n) = (-1)(f(n-2) + 2f(n-1)). Prove that f(n) = (1-2n)(-1)^n.
- Suppose G is a connected graph with only even vertices.
Prove that none of the edges in G is a bridge.
- Suppose G is a graph, and u,v,and w are vertices. Prove
that if u is connected to v and v is connected to w then u is connected
to w. Is this true if we replace connected
with adjacent?
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Due Feb. 19
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Practice problems:
- Section 1.2, #6,12, 18
- Section 2.2, # 5, 7, 18,19
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Hand in:
- Section 1.2, #19
- Suppose you have tons of 5-cent and 7-cent stamps. What
values of postage that are less than 25 cents are impossible to get
using only these stamps? Prove that if p > 23, then there is a
combination of 5-cent and 7-cent stamps that adds to p. Use strong
induction
- Let G be the graph whose vertices are
{1,2,...,14,15}. Two vertices are adjacent if and only if
they have a common factor which is greater than 1. Draw a picture of G.
How many connected components does it have? What is the length of a
longest path in G? A longest cycle?
- Section 2.2, #16
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Due Feb. 16
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Practice problems:
- Section 1.1, #6, 20, 23, 25 - 29
- Section 2.1, #3, 4, 8
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Hand in:
- Section 1.1, #34, 39
- Section 2.1, # 6
- Prove that if n and k are both odd integers, then there is
no k-regular graph on n vertices.
- Recall that a rational number is simply a fraction of the
form a/b, where a and b are integers and b is not zero. Prove that if r
and s are rational numbers and s is non-zero, then r/s is an integer.
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