21 problems

Let $S=K[x_1,\ldots,x_n]$, let $J_{n,m}$ be the path ideal of the cycle graph on $n$ vertices with paths of length $m$, and set $d=\gcd(n,m)$. Let $t_0\leq n-1$ be maximal such tha…

Let $C_n$ be the cycle graph on $n$ vertices, let $C_n^p$ be its $p$-th power, and let $\Delta_3(C_n^p)$ denote its 3-cut complex. Shellability conjecture. The complex…

Let $C_n$ be the cycle graph on $n$ vertices, let $C_n^2$ be its square, and let $\Delta_k(C_n^2)$ denote its $k$-cut complex. Bayer et al.'s conjecture. Motivated by computational…

Let $C_k$ denote the cycle graph on $k$ vertices, and let $\mathrm{sat}(n,C_k)$ be its saturation number. Füredi–Kim conjecture. There exists a constant $k_0$ such that … holds for…

Let $C_n^r$ denote the $r$th power of the cycle on $n$ vertices, and let $\Delta_d^t(C_n^r)$ denote its total $d$-cut complex. For $d\geq3$ and $n\geq(2d-1)r+1$, Shen et al.'s conj…

Cyclic interval conjecture. The displayed inequality holds for every odd $m$ and every nonempty proper subset $x\subseteq\mathbb{Z}_m$; consequently,

Let…

Let $C_6$ be the $6$-cycle, let $C_6(t)$ denote the associated $3$-uniform hypergraph obtained by adjoining a new vertex to every edge of $C_6$, and let $\operatorname{ex}(n,H)$ de…

Let $C_n$ be the cycle graph on $n$ vertices, and let $J_{C_n}$ denote its binomial edge ideal. The v-number of a homogeneous ideal is the minimum degree of a homogeneous polynomia…

Dey's conjecture. For $n\geq 6$, equality holds in the bound

Online cycle Ramsey conjecture.

Let $C_8$ be the cycle on eight vertices, and let $\operatorname{ex}(n,H)$ denote the maximum number of edges in an $H$-free graph on $n$ vertices. $C_8$ extremal-number conjecture…

Let $C$ be the maximum constant such that, for every $0<\varepsilon<C$ and all sufficiently large $n$, every $n$-vertex graph $G$ with $\lambda(G)>\sqrt{\lfloor n^2/4\rfloor}$ cont…

Spectral cycle-length conjecture. The largest such constant $C$ is characterized by the property that every such graph contains a cycle of every length $t\le (C-\varepsilon)n$. The…

Induced 4-cycle construction conjecture. For every real number $x\in[1/2,1]$,

The anti-directed cycle Turán conjecture. For every $k\in \mathbb{N}^*$, there exists a constant $n_0$ such that for $n\ge n_0$,

Let $C_4$ denote the cycle of length four, and let $\operatorname{ex}^{-1}(k,H)$ denote the inverse Turán number of $H$, namely the maximum number of edges in a graph whose every…

Let $C_t$ be the cycle on $t$ vertices, and let $f_{C_t}(s,s)$ denote the least number of independent sets of size $s$ in $C_t$ that guarantees a rainbow independent set of size…

Cycle-union conjecture. The union

Deretsky–Lee–Miller conjecture. Every union of cycles with at most one odd cycle has a prime labeling.

Nonexistence conjecture. For $t\ge 6$, there are no nontrivial uniquely $C_t$-saturated graphs.