18 problems

Let $\Gamma$ be a graph, let $\delta>2$, and let a transition system of $\Gamma$ be the structure pairing incident edges as used in the paper. An edge cut is non-trivial when it do…

A bipartite Eulerian tournament is an orientation of a complete bipartite graph in which every vertex has equal indegree and outdegree. Let $C_4$ denote the directed cycle on four…

Let $G$ be an Eulerian graph, and call an integer feasible if $G$ has a cycle decomposition containing that many odd cycles. Parity interval conjecture. If $r$ and $t$ are feasible…

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…

Let $D$ be a directed Eulerian graph, meaning a digraph in which every vertex satisfies $d^+(v)=d^-(v)$. A cycle decomposition partitions the directed edges into directed cycles. D…

Let $q\geq 2$ be an integer, let $G$ be a graph of order … where $n_i\geq 5$ for each $i\in [q]$, and let $\{f_1,\ldots,f_q\}$ be $q$ vertex-disjoint arcs in $G$. Wang's prescribed…

Let $D$ be a directed graph. For a cut of $D$, consider the cardinalities of the edges crossing the cut in each of the two directions. Thomassen's conjecture. The edges of $D$ can…

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 $H$ be a $3$-uniform hypergraph, let $|H|$ denote its number of vertices, and let $[?]$ be its minimum codegree. A cycle decomposition is a decomposition of the edges of $H$ in…

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 $G$ be a directed Eulerian graph on $n$ vertices, meaning that each vertex has equal indegree and outdegree. Linear cycle decomposition conjecture. Every directed Eulerian grap…

Mahmoodian–Mirzakhani conjecture. These necessary conditions are also sufficient for $K_{r,s,t}$ to decompose into 5-cycles. This conjecture gives necessary and sufficient conditio…

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…

Let $G$ be a graph, and consider the minimum degree threshold guaranteeing that $G$ has a decomposition into cycles of length six. The preceding results establish an asymptotic thr…

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 $K_5$.