14 problems

Let $r\in\mathbb{N}$, and let the edges of a complete graph be coloured with $r$ colours. A monochromatic cycle partition is a partition of the vertex set into vertex sets, each in…

Let $G$ be a graph, let $r$ be a positive integer, and let $\alpha(G)$ denote the independence number of $G$. For an $r$-edge-colouring of $G$, let $\operatorname{cp}_r(G)$ be the…

Let $k\geq 2$ and consider a finite complete $k$-uniform hypergraph, meaning that every $k$-element subset of its vertex set is an edge. An edge-colouring assigns a colour to each…

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…

Let $G$ be an $n$-vertex graph whose edges are coloured with two colours. Balogh–Barát–Gerbner–Gyárfás–Sárközy conjecture. If the minimum degree of $G$ is greater than $3n/4$, then…

Let $G$ be an $n$-vertex graph whose edges are coloured red and blue, and let $\deg(x)$ denote the degree of a vertex $x$. Barát–Sárközy's conjecture. If … for every pair of non-ad…

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 $G$ 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…

Linear cycle-cover conjecture. There is a $c$ such that for every $r$, every $r$-local colouring of $K_n$ admits a covering with $cr$ disjoint monochromatic cycles. The same should…

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…