16 problems

Average-degree rainbow path conjecture. If $G$ has average degree at least $d$, then every proper edge-coloring of $G$ contains a rainbow copy of the path on $d$ edges.

Let $G$ be a strongly edge-colored graph on $n$ vertices, and let $\delta(G)$ denote its minimum degree. Cheng, Sun, Tan and Wang's conjecture. If … then $G$ has a rainbow Hamilton…

Rainbow triangle-packing conjecture. The graph $K_n$ contains a rainbow subgraph that is a disjoint union of triangles covering all but at most $C$ vertices, for some absolute cons…

Ryser's conjecture. If $n$ is odd, then $\vec{K}_n$ contains a rainbow spanning subgraph in which every vertex has in-degree and out-degree equal to one. Equivalently, $\vec{K}_n$…

Let $M$ be a simple regular matroid of rank $n-1$, and let $c$ colour $E(M)$ with $n$ colours, each colour class having size at least $2$. Regular matroid rainbow circuit conjectur…

Rainbow Turán path conjecture. For all $\ell \geq 3$,

Let $\gamma$ be the maximum on $[0,1]$ of … and suppose that the maximum is attained at $x=x_0$. Consider three graphs $G_1,G_2,G_3$ on a common vertex set of size $n$, with no rai…

Let $G_1,G_2,G_3$ be graphs with a common vertex set on $n$ vertices. A rainbow triangle is a triangle whose three edges belong to three distinct graphs among $G_1,G_2,G_3$. Frankl…

A graph is properly colored if adjacent vertices receive distinct colors, and a vertex-colored graph is rainbow if no two vertices have the same color. A graph is triangle-free if…

Rainbow bandwidth conjecture. There are $\nu,\beta,z>0$ and $n_0\in\mathbb{N}$ such that, whenever $G$ additionally has an edge-colouring in which every colour appears on at most…

Let $K_n$ be a complete graph with a proper edge-colouring, meaning that any two incident edges have different colours. A subgraph is rainbow if all its edges have distinct colours…

Let $K_n$ be a complete graph and let a $1$-factorization be an edge-colouring whose colour classes form a decomposition of $K_n$ into perfect matchings. A subgraph is rainbow if a…

Let $K_n$ be a complete graph and let a $1$-factorization be an edge-colouring whose colour classes form a decomposition of $K_n$ into perfect matchings. A subgraph is rainbow if a…

Let $K_n$ be the complete graph on $n$ vertices. An edge-colouring is $k$-bounded if no colour appears on more than $k$ edges. A subgraph is rainbow if no two of its edges have the…

Let $\varepsilon>0$. A graph is globally $m$-bounded if no colour is used on more than $m$ edges, and a forest of order $k$ has $k$ edges. Rainbow forest packing conjecture. There…

For positive integers $n$ and $t$, let $r(n,t)$ be the maximum number of colors in an edge-coloring of $K_n$ that has no $t$ edge-disjoint rainbow spanning trees. Jahanbekam–West c…