# 6 1 First, write down a list of your daily activities that you typic

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6.1 First, write down a list of your daily activities that you
typically do on a weekday. For instance, you might get out
of bed, take a shower, get dressed, eat breakfast, dry your
so you have a minimum of 10 activities.
6.1.1 Now consider which of these activities is already
exploiting some form of parallelism (e.g., brushing multiple
teeth at the same time, versus one at a time, carrying one
book at a time to school, versus loading them all into your
backpack and then carry them \"in parallel\"). For each of
parallel, but if not, why are not.
6.1.2 Next, consider which of the activities could be carried
out concurrently (e.g., eating breakfast and listening to the
news). For each of your activities describe which other
activity could be paired with this activity.
6.1.3 For 6.1.2, What could we change about current
system (e.g., showers clothes, TVs, cars) so that we could
Solution
For a subset of a finite group , the Cayley digraph is the
directed graph whose vertices are the elements of and
with a directed edge for every and . The
correspondingCayley graph is the underlying undirected
graph that is obtained by removing the orientations from all

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the directed edges.
It has been conjectured that every (nontrivial) connected
Cayley graph has a hamiltonian cycle. (See the
bibliography of [1] for some of the literature on this
problem.) This conjecture does not extend to the directed
case, because there are many examples of connected
Cayley digraphs that do not have hamiltonian cycles. In
fact, infinitely many Cayley digraphs do not even have a
hamiltonian path.
Proposition 2 (attributed to Milnor [2, p. 201]). Assume the
finite group is generated by two elements and , such that
. If , then the Cayley digraph does not have a hamiltonian
path.
The examples in the above proposition are very
constrained, because the order of one generator must be
exactly 2 and the order of the other generator must be
exactly 3. In this note, we provide an infinite family of
examples in which the orders of the generators are not
restricted in this way. In fact, and can both be of
arbitrarily large order.
Theorem 3. For any , there is a connected Cayley digraph ,
such that(1) does not have a hamiltonian path,(2)
and both have order greater than .Furthermore, if is any
prime number such that and , then we may construct the
example so that the commutator subgroup of has order .

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6.1 First, write down a list of your daily activities that you typically do on a weekday. For instance, you might get out of bed, take a shower, get dressed, eat breakfast, dry your hair, brush your teeth. Make sure to break down your list so you have a minimum of 10 activities. 6.1.1 Now consider which of these activities is already exploiting some form of parallelism (e.g., brushing multiple teeth at the same time, versus one at a time, carrying one book at a time to school, versus loading them a ll into your backpack and then carry them \"in parallel\"). For each of your activities, discuss if they are already working in parallel, but if not, why are not. 6.1.2 Next, consider which of the activities could be carried out concurrently (e.g., eating breakfast and listening to the news). For each of your activities describe which other activity could be paired with this activity. 6.1.3 For 6.1.2, What could we change about current system (e.g., showers clothes, TVs, cars) so that we could perform more tasks in parallel? Solution For a subset of a finite group , the Cayley digraph is the directed graph whose vertices are the elements of and with a directed edge for every and . The correspondingCayley graph is the underlying undirected graph that is obtained by removing the orientations from all the directed edges. It has been conjectured that every (nontrivial) connected Cayley graph has a hamiltonian cycle. (See the bibliography of [1] for some of the literature on this pr ...
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