Question

MATH 353

2. (a) For which values of n > 1, if any, is K_{n} an
Euler graph? Explain.

(b) For which values of n > 1, if any, does K_{n}
have an Euler path? Explain.

Answer #1

For which values of n ≥ 3 do these graphs have an Euler
cycle?
(a) Complete graph Kn
(b) Cycle graph Cn
(c) Complete bipartite graph Kn,n

Discrete math
Use mathematical induction to prove that n(n+5) is divisible by
2 for any positive integer n.

(-) Prove that 1·2 + 2·3 +···+ (n−1) n = (n−1)n(n+ 1) /3.
(Discrete Math - Mathematical Induction)

consider
1/n. Find all the values of n for which 1/n has a decimal
representation with period 1. You may use the fact that numbers of
the form n=2^a5^b have a period of 1 without having to prove
it

consider 1/n. Find all the values of n for which 1/n has a
decimal representation with period 1. You may use the fact that
numbers of the form n=2^a5^b have a period of 1 without having to
prove it.

11.43. Let 1 ≤ m ≤ n. Show that
(a). Km ∩ Kn =
Km,
(b). Km ∪ Kn =
Kn,
(c). Km ⊆ Kn.

a. A graph is called k-regular if every vertex has degree k.
Explain why any k-regular graph with 11 vertices must conain an
Euler circuit. (Hint – think of the possible values of k). b. If G
is a 6-regular graph, what is the chromatic number of G? Must it be
6? Explain

2.) Consider an input array A of size n in which n − 1 of the
elements have identical values and the remaining one value is
smaller than the n − 1 identical values. What is the running time
of Heapsort with input A?

1a. Find any of the values a1, r, an, n, or sn that are missing
from the geometric sequence
r= 5/2, n=4, S4=1624
the value of a1 is?
b. find the sum of the first n terms of the indicated geometric
sequence with the given values
1/8 + 1/2 + 2 + (X) times (X) times (X)times + 32

For what values of m, n ∈ Z>0 does the complete bipartite
graph Km,n have a perfect matching? Prove it

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