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

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 all 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

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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