18 problems

Let $n\in\mathbb{N}$ with $n\geq 3$, and let $\mathcal{G}=\{G_i:i\in[n]\}$ be a family of $n$-vertex graphs with the same vertex set; the graphs in the family may be identical. For…

Let $n$ be even. In the rainbow perfect matching game $\mathcal{RP}_n$, played on $n/2$ copies of $K_n$, Maker wins by claiming a rainbow perfect matching; let $b_{\mathcal{RP}_n}$…

Let $\mathcal{RS}_n$ be the rainbow-spanning-tree game played on $n-1$ copies of $K_n$, where Maker wins by claiming a rainbow spanning tree, and let $b_{\mathcal{RS}_n}$ denote it…

For $X\subseteq\mathbb{N}$, let $K_X$ be the complete graph on vertex set $X$ in which each edge $ij$ has colour $|i-j|$. Write $[n]=\{1,\ldots,n\}$. A copy of a tree in this edge-…

Equivalent Rainbow Arborescence Conjecture. If $k \ge n-1$, then the disjoint union $G$ of $k$ spanning arborescences has a rainbow spanning arborescence $B$.

Yokoi's Rainbow Arborescence Conjecture. If $G$ is the disjoint union of $n-1$ spanning arborescences $A_1, A_2, \dots, A_{n-1}$, then $G$ has a rainbow spanning arborescence $B$.

Let $G$ be a Dirac graph on $n$ vertices, meaning a graph with minimum degree at least $n/2$, and let its edge-colouring be proper if every pair of incident edges receives differen…

Let $G$ be a Dirac graph on $n$ vertices, meaning a graph with minimum degree at least $n/2$, and let its edge-colouring be proper if every pair of incident edges receives differen…

Let $\mathcal{G} = \{G_1, \ldots, G_n\}$ be a collection of graphs with common vertex set $V$ of size $n$. Define $\delta(\mathcal{G})=\min\{\delta(G_i):1\leq i\leq n\}$, where…

Let $K_n$ be the complete graph on $n$ vertices, and let $\chi'(K_n)$ denote its edge-chromatic number. A properly edge-coloured graph is one in which adjacent edges receive distin…

Let $k,c,c'\in\mathbb{N}$ with $c'\leq c$, and let $G$ be a graph. For $r\in\{c,c'\}$, let $T(r,k)$ be the random variable counting the number of rainbow trees of order $k$ when th…

Let $G_c(n,p)$ be the random graph on $n$ vertices whose edges are independently present with probability $p$ and uniformly coloured with $c$ colours. Let $b1>0$, set $c=b1 n$, let…

Let $M$ be a $k$-base matroid, and let $\mathcal{P}$ be a 2-bounded partition of $E(M)$, meaning that every partition class has size at most $2$. Rainbow basis covering conjecture.…

Let $M$ be a $k$-base matroid with $k\geq 3$, and let $\mathcal{P}$ be a partition of $E(M)$ satisfying … for every $X\in\mathcal{P}$. 2-bounded rainbow factorization conjecture. T…

Let $M$ be a $k$-base matroid, meaning that its ground set can be partitioned into $k$ bases, and let $\mathcal{P}$ be a partition of $E(M)$ such that … for every $X\in\mathcal{P}$…

Let $\overleftrightarrow{K_n}$ be the digraph obtained from the complete graph $K_n$ by replacing each edge with two oppositely directed arcs and adding a directed loop at each ver…

Aharoni–Barat–Wanless conjecture. The graph $G$ has a rainbow matching using every colour.

Let $n$ be a positive integer, let $p$ satisfy … and let $\mathcal{G}_c(K_{n,n},p)$ denote the random graph on the complete bipartite graph $K_{n,n}$ whose present edges receive a…