12 problems

Let $n$ be an even integer with $n\ge 4$. A $1$-factorisation of $K_n$ is a partition of its edge set into perfect matchings; two perfect matchings are called perfect when their un…

Let $K_{2n}$ be the complete graph on $2n$ vertices. A perfect matching is a spanning 1-regular subgraph, and a Hamiltonian cycle is a cycle containing every vertex. Perfect 1-fact…

Let $Q_d$ be the $d$-dimensional hypercube. A 1-factorisation is semi-perfect if one 1-factor has the property that its union with every other 1-factor is a Hamilton cycle. Craft's…

Let $Q_d$ be the $d$-dimensional hypercube, and let $alM$ be a 1-factorisation of $Q_d$ sampled uniformly from the set of all such 1-factorisations. Define $r(alM)$ to be the minim…

Equitable-factorization characterization conjecture. If $G$ is a $k$-tree-connected graph, then $G$ admits a $k$-equitable factorization if and only if, for every…

Almost equitable factorization conjecture. The graph $G$ can be edge-decomposed into factors $G_1,\ldots,G_k$ such that, for every $v\in V(G)$,

Let $\mathcal{U}$ be the Latin square representing a uniformly random 1-factorisation of $K_{2n}$. Let $\ell(\mathcal{U})$ be the length of its longest row cycle, and let…

For integers $n\geq1$, let $f_2(3,n)$ be the minimum total number of maximal cliques over all factorizations of the complete graph $K_n$ into three edge-disjoint spanning subgraphs…

For $d>2$, let $Q_d$ be the hypercube graph. A perfect 1-factorization is a partition of the edges of $Q_d$ into perfect matchings such that the union of every two distinct matchin…

Let $i$ and $n$ be such that $(i+1)\mid(n+1)$, and let the $i$-spaces of $\mathrm{PG}(n,q)$ be the relevant subspaces. Subspace-induced factorization conjecture. There exists a…

An elliptic quadric is a cap in $\mathrm{PG}(3,q)$, and $\mathrm{PG}(3,q)$ can be partitioned into $q+1$ disjoint elliptic quadrics. The set of elliptic quadrics induces factors on…

An $M$-cap in $\mathrm{PG}(n,q)$ is a set of $M$ points with no three collinear. By Ebert's theorem, if $n\geq 2$ is even, then $\mathrm{PG}(2n-1,q)$ can be partitioned into disjoi…