Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Pancyclicity when each cycle contains k chords   For integers $n \geq k \geq 2$, let $c(n,k)$ be the minimum number of chords
that must be added to a cycle of length $n$ so that the resulting graph has the
property that for every $l \in \{ k , k + 1 , \dots , n \}$, there is a cycle
of length $l$ that contains exactly $k$ of the added chords. Affif Chaouche,
Rutherford, and Whitty introduced the function $c(n,k)$. They showed that for
every integer $k \geq 2$, $c(n , k ) \geq \Omega_k ( n^{1/k} )$ and they asked
if $n^{1/k}$ gives the correct order of magnitude of $c(n, k)$ for $k \geq 2$.
Our main theorem answers this question as we prove that for every integer $k
\geq 2$, and for sufficiently large $n$, $c(n , k) \leq k \lceil n^{1/k} \rceil
+ k^2$. This upper bound, together with the lower bound of Affif Chaouche et.\
al., shows that the order of magnitude of $c(n,k)$ is $n^{1/k}$.

7
The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ