7 problems

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…

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…

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…

Let $STS(n)$ be the number of Steiner triple systems on $n$ points and let $F(n)$ be the number of 1-factorizations of the complete graph $K_n$. Wilson's joint asymptotic conjectur…