Given a $d$-regular graph with $n$ vertices is there a known (non-trivial) upper bound on the length of [chordless cycles](http://mathworld.wolfram.com/ChordlessCycle.html) in it (presumably as a function of $d$ and $n$)? I wasn't able to find anything after some online searches. Thank you.



Upper bound on the length of chordless cycles in d-regular graphs
Say a chordless cycle $C$ has length $k$.
The number of edges between $C$ and $V-C$ is
$k(d-2)$. Also, the number of edges from $V-C$ to $C$ is at most $(n-k)d$. The solution is
$$k\le \biggl\lfloor
 \frac{nd}{2(d-1)}\biggr\rfloor.$$
This is often realised but I'm too lazy to figure out if it is always realised.


