16 problems

Verstraëte's conjecture. If $G$ does not contain $k$ cycles of consecutive even lengths, then

Let $G$ be a $k$-regular Hamiltonian graph on $n$ vertices. Jacobson–Lehel conjecture. For every $k\geq 3$, $G$ contains cycles of at least $\Omega(n)$ distinct lengths; equivalent…

Liu and Ma's conjecture. Every graph with minimum degree at least $k+1$ contains $k$ admissible cycles.

Diwan–Kenkre–Vishwanathan conjecture. If a graph $G$ has no cycle of length $m$ modulo $k$, then its chromatic number satisfies

Asymptotic excess conjecture.

Conjecture on consecutive cycle lengths. Except when $G$ is $K_{k+1}$, the graph $G$ contains $k$ cycles of consecutive lengths.

Liu–Ma conjecture. $G$ contains $eil k/2$ cycles with consecutive odd lengths.

Let $G$ be an $n$-vertex Hamiltonian graph with minimum degree at least $3$. Concentration and separation conjecture. Both of the following hold: 1. $G$ has $\Omega(\sqrt n)$ cycle…

Let $G$ be an $n$-vertex Hamiltonian graph, and let $\delta(G)$ denote its minimum degree. Verstraëte's conjecture. If … then $G$ has $\Omega(n)$ different cycle lengths. This stre…

Let $G$ be a graph with girth $g$ and average degree $d$, and let $\mathcal{C}(G)$ denote its cycle spectrum, the set of cycle lengths occurring in $G$. Erdős conjecture. One has ……

Let $f_2(n)$ denote the maximum, over all $n$-vertex 2-connected graphs, of the number of cycle lengths that occur exactly once. The authors' conjecture is … This conjecture assert…

For $k\geq 2$, let $\chi_k$ be the largest chromatic number of a graph that does not contain $k$ cycles of consecutive lengths. Sudakov–Verstraëte's conjecture. For every integer…

Conjecture on consecutive odd cycle lengths. If $G$ is a 2-connected non-bipartite graph with minimum degree at least $k+1$, then $G$ contains $\lceil k/2\rceil$ cycles with consec…

A degree 3-critical graph is a graph on $n$ vertices with $2n-2$ edges and no proper induced subgraph of minimum degree $3$. Cycle-count conjecture. Every degree $3$-critical graph…

Uniform splitting conjecture. For every $\varepsilon>0$,

Let $T$ satisfy $uinfty(T)0$, and let $R{ij}$ and $Ri$ denote the rates at which distinct macroscopic cycles of indices $i,j$ merge and the macroscopic cycle of index $i$ splits,…