10 problems

Hajós' conjecture. Every simple Eulerian graph $G$ has a cycle decomposition with at most

Odd-cycle decomposition threshold conjecture.

Let $G$ be an $n$-vertex $(2k+1)$-regular graph, where $k$ is a nonnegative integer. Let $\operatorname{f_{re}}(G)$ denote the minimum number of $2$-regular graphs and edges in a c…

Let $G$ be an $n$-vertex graph. A cycle-and-edge cover is a cover of the edge set of $G$ by subgraphs that are $2$-regular graphs or single edges. The linear cycle-and-edge cover c…

Even-cycle equitable-colourability conjecture. There exists an equitably 2-colourable $\ell$-cycle decomposition of $K_v-I$ if and only if

Let $k$ be an integer with $k\geq 3$. For a $k$-graph $H$, a cycle-decomposition is an edge partition of $H$ into tight cycles, and the cycle-decomposition threshold…

Let $G$ be an Eulerian directed graph on $n$ vertices, meaning that its indegree equals its outdegree at every vertex. Bollobás–Scott partition conjecture. The edge set of $G$ can…

Let $K_n^h$ denote the complete $h$-uniform hypergraph on $n$ vertices, and let $bb K_n^h$ be the hypergraph with $bb$ copies of every edge. A regular Hamiltonian cycle is a Hamilt…

A signed graph is a graph whose edges are designated even or odd; a cycle is even when it contains an even number of odd edges. A graph is even cycle decomposable if its edge set c…

Fleischner's conjecture. The pair $(G,T)$ has no compatible cycle decomposition if and only if $(G,T)$ is the bad loop or the bad $K_5$.