# Cayley graphs and standard sets

What does "The cayley graph on every semidirect product of two cyclic groups with standard generating set has a hamiltonian cycle" mean?

There some other conditions satisfied by the above statement but, what does "standard set" mean? Is it the generator set considered to draw the cayley graph? If it is so how do I find a standard set like that?

You are right that this part of the sentence is trying to tell you which generating set to use to construct the Cayley graph of the group.

It's telling you: use "the standard one". In this case, "standard" is not mathematical terminology, but just means that we do the most obvious thing.

What the most obvious thing is might not necessarily be all that obvious. In this case, I would say that

1. The standard generating set of a cyclic group is a single element generating the whole group.
2. The standard generating set of the semidirect product of two groups is the union of the elements generating each group (viewed as subgroups of the semidirect product).

So in this case, the generating set you are asked to use consists of two elements: an element generating the first cyclic group, and an element generating the second.

• Thank you very much. Will the Cayley graph obtained with the standard generating set be a subgraph of all other Cayley graphs constructed using any other generating set? Can we conclude something like that? – Buddhini Angelika Aug 27 '18 at 4:40
• Not necessarily. The most you can say is that if one generating set is a subset of another, then the Cayley graph for the first will be a subgraph of the Cayley graph for the second. But consider, for example, the Cayley graphs of $\mathbb Z$ using the generating sets $\{1\}$ and $\{2,3\}$; neither is a subgraph of the other. – Misha Lavrov Aug 27 '18 at 4:49
• How about if it is a semidirect product as mentioned above? If it is a semidirect product of finite cyclic groups – Buddhini Angelika Aug 27 '18 at 5:00
• Same thing holds there; there is no relationship in general. – Misha Lavrov Aug 27 '18 at 5:02
• Ok, thank you very much. – Buddhini Angelika Aug 27 '18 at 5:12