11

On the Wikipedia page of the Petersen graph it is mentioned that it is not a Cayley graph.

enter image description here

How it this proved?
Honestly I don't even know how to start this. The only criteria I can think of is that all vertices must have the same degree. Also the degree being odd should imply that one generator of the group has order 2. But then how do I proceed?

Edit: As already mentioned in the comments, the link did not answer my question.

stacky
  • 165
  • I guess I shouldve used the search function – stacky Jul 16 '18 at 16:50
  • It's a good question, if already asked: https://math.stackexchange.com/questions/348779/petersen-graph-is-not-a-cayley-graph - although the answer there could be improved. Basically it can only correspond to a small group, of which there are a limited number, and it doesn't so correspond. – Joffan Jul 16 '18 at 16:55
  • 1
    @Joffan I took a look at the answer, but I dont really get it. It seems like a fact from group theory (which i have to accept) that there are only two groups with $10$ elements, These are $D_5$ and $\mathbb Z_10$. It is clear to me that for the standard generating sets of these groups, the Cayley graphs differ. How can I see that there is NO generating set realising the graph? – stacky Jul 16 '18 at 16:58
  • $C_{10}$ is cyclic so clearly not appropriate. $D_{5}$ has an order-2 element which should lead to even cycles on the graph, but Petersen has only odd (unlike say the pentagonal prism graph). – Joffan Jul 16 '18 at 17:09
  • @Joffan ah, I see, in a different generating set the order 2 element would be a $n$-word. This would then give a $2n$-cycle. Why is the case $C_{10}$ "clear"? – stacky Jul 16 '18 at 17:15
  • Aren't Cayley graphs always regular of even degree? – hmakholm left over Monica Jul 16 '18 at 17:20
  • @HenningMakholm No, check the Cayley graph of $\mathbb Z_2$ – stacky Jul 16 '18 at 17:21
  • @stacky: Ah, right. – hmakholm left over Monica Jul 16 '18 at 17:22
  • @Joffan There are certainly cycles of length $6$ in the Petersen graph. – Derek Holt Jul 16 '18 at 17:23
  • @DerekHolt true... maybe there isn't such a simple argument. – Joffan Jul 16 '18 at 17:27

1 Answers1

14

$G = C_{10}$ is abelian, and so if $a$ and $b$ are two of the generators in the Cayley graph then $a^{-1}b^{-1}ab$ gives a cycle of length $4$, but the Petersen graph has none such.

So suppose that $G= D_{10}$. As you say, at least one of the generators, say $a$ must have order $2$. If there is a generator $b$ of order $5$, then $(ab)^2 = 1$, so again we should have a cycle of length $4$, but there is none.

The only other possibility is that all generators have order $2$, but then there is no product of $5$ generators equal to the identity. But the Petrsen graph does have $5$-cycles, contradiction.

Derek Holt
  • 96,726
  • Thx for your answer. What do you mean with "the other possibility". Do you refer to another group with 10 elements in this part?

    Also, why do you say "at least one of the generators must have order 2". Its clear that there are elements of order 2, but why must I take such an element as a generator?

     – stacky Jul 16 '18 at 17:24
  • The third paragraph is still assuming $G = D_{10}$ (which is confusingly often denoted by $D_5$). By "the only other possibility" I meant that if there is no generator of order $5$, then all of the generators must have order $2$. As you said yourself, the fact that the degree is odd implies that at least one of the generators has order $2$. – Derek Holt Jul 16 '18 at 17:28
  • Sorry, but why is $(ab)^2=1$? – stacky Jul 16 '18 at 17:36
  • In dihedral groups the product of a rotation and a reflection is a reflection. – Derek Holt Jul 16 '18 at 18:10
  • ah, it comes from the explicit group structure. Thx – stacky Jul 16 '18 at 18:20