13 problems

Let $d_1\leq\dots\leq d_n$ be the degree sequence of a graph $G$. The degree-sequence conjecture. There is a function $f$ such that, for every $\beta>0$, every integer $r\geq 2$, a…

For $s,k\in\mathbb{N}$ with $k\geq 2$, let $c=c(s,k)\in\mathbb{N}$ be a constant independent of $n$, and let $\mathcal{K}_{n}^{(k)}$ denote the complete $k$-uniform hypergraph on…

Let $k$ and $r$ be positive integers. A $k$-uniform hypergraph is a hypergraph whose edges are $k$-element subsets of its vertex set, and a tight cycle is a cyclically ordered sequ…

Let $r$ be a positive integer, and let a complete graph have its edges coloured with $r$ colours. A collection of vertex-disjoint monochromatic cycles consists of cycles with pairw…

Let $r$ be a positive integer, and let an $r$-edge-coloured complete graph be a complete graph whose edges receive one of $r$ colours. Erdős–Gyárfás–Pyber conjecture. The vertices…

Let $G$ be a graph, and let $\overline G$ denote its complement. Lehel's conjecture. The vertices of $G$ can be covered by a cycle $C_1$ in $G$ and a vertex-disjoint cycle $C_2$ in…

Let $G$ be a $t$-colored graph with independence number $\alpha(G)=\alpha$, and let $n=|V(G)|$. Asymptotic Sárközy cycle-cover conjecture. There exists a constant $c=c(\alpha,t)$ s…

Let $G$ be a $t$-colored graph, and let $\alpha(G)$ denote its independence number. Let the cycle partition number be the minimum number of vertex-disjoint monochromatic cycles cov…

Erdős–Gould–Yuster–P conjecture. The cycle partition number of any $r$-colored complete graph is at most $r$.

Let $K_n$ be a complete graph whose edges are coloured with $r$ colours. A cycle covering is a collection of $r$ monochromatic cycles whose union of vertex sets contains all vertic…

Let $K_n$ be a complete graph whose edges are coloured with $r$ colours. A vertex-disjoint monochromatic cycle packing is a collection of $r$ vertex-disjoint monochromatic cycles.…

Linear-length monochromatic cycle conjecture. If

Monochromatic cycle interval conjecture. If $n\geq4$ and $\delta(G)>3n/4$, then every $2$-coloring of $E(G)$ has, for every $k\in[4,\lceil n/2\rceil]$, either a red $k$-cycle or a…