# Is there a name for this: an even cycle having two smaller even cycles inscribed, linking bipartitions

My apologies for the horribly worded title, I'm not sure how to describe this.

Learning about bipartite graphs in discrete math and I noticed this when testing a few cycle graphs. For $n$ even, $C_n$ can have two graphs $C_{n/2}$ inscribed within, which connect the two bipartitioned sets.

See examples here for $C_6$, $C_8$, and $C_{10}$. (the last one is hard to see, but it does have two pentagons inscribed within it) So $C_6$ has a $C_3$ that connects the nodes of one of its bipartitioned sets, and another $C_3$ connecting the nodes in the other set.

Is there a name for this property/feature/whatever?

• I'm not sure I understand what property/feature you are referring to. As you have noted, the even cycle $C_{2k}$ is bipartite, which each part being an independent set of size $k$. Hence you could fit a cycle $C_k$ into each part, but you could also fit any graph on $k$ vertices into each part. Jul 4, 2016 at 21:19
• @Shagnik thanks for the feedback, I just wasn't sure how to word the question. I was focusing on cycles specifically and it just seemed interesting that we can nest a cycle of exactly half size, so I was looking for some deeper connection. Which is obvious now that I look back on it, because that is just constructing a cycle for each partition, of which there are two.
– Dave
Jul 4, 2016 at 22:06