20 problems

Let $G$ be an $n$-vertex edge-colored graph, where the minimum color degree is $4\delta^c(G)=\min_{v\in V(G)}d^c_G(v)$, with $d^c_G(v)$ the number of distinct colors on edges incid…

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…

Let $K_n$ be the complete graph on $n$ vertices, and let a proper edge-colouring be an edge-colouring in which adjacent edges receive different colours. A rainbow spanning tree is…

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, with an edge-coloring that is $k$-bounded when no color appears on more than $k$ edges. A rainbow Hamiltonian cycle is a Hamiltonia…

Let $G_1,G_2,G_3\subset\binom{[n]}{2}$ be simple graphs whose edge-colored multiple graph is rainbow triangle-free, meaning that it contains no triangle whose three edges have dist…

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…

Let $n$ and $k$ be positive integers with $n\geq 3k$. For an edge-colored graph $G$ of order $n$, let $\delta^c(G)$ denote its minimum color degree, and call a triangle rainbow whe…

Let $T$ be a tree on at most $(1-\varepsilon)n$ vertices with maximum degree at most $d$, where $\varepsilon\in(0,1)$ and $d\geq 2$ is an integer. Consider a uniformly coloured ran…

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…

Let $H_1,\ldots,H_\ell$ be the almost edge-disjoint rainbow subgraphs of excess $k$ produced by the paper's many non-star case. Short rainbow cycle conjecture for the constructed s…

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…

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…

Generalized rainbow Turán conjecture. For odd $k$, this upper bound is tight, so

Let $n$ be the number of vertices and let $C_{2k}$ be the cycle of length $2k$. Write $\operatorname{ex}^*(n,C_{2k})=\operatorname{ex}(n,P_1,\textnormal{rainbow-}C_{2k})$ for the m…

An edge-coloured graph is $k$-bounded if no colour appears more than $k$ times. A rainbow Hamiltonian cycle is a Hamiltonian cycle whose edges have pairwise distinct colours. Hahn…

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…

Let $K_{2n}$ have a $1$-factorization, meaning a proper edge-colouring with $2n-1$ colours whose colour classes are perfect matchings. A subgraph is rainbow if all its edges have d…