8 problems

Let $G$ be a graph of even order $n$, and suppose that $G$ is $k$-regular for some integer $k\ge 2\left\lceil\frac{1}{4}n\right\rceil-1$. 1-Factorization Conjecture. $G$ is 1-facto…

Let $K_{2n}$ be the complete graph on $2n$ vertices. A 1-factorization is a decomposition of its edges into perfect matchings, and it is perfect when the union of any two distinct…

Let $G$ be an $N$-regular graph with $2n$ vertices. A 1-factorization is a decomposition of the edges of $G$ into perfect matchings. The 1-factorization conjecture. If $n$ is odd a…

Let $K_{4n+1}$ be a complete graph on $4n+1$ vertices, where … and both $m_1$ and $m_2$ are even. A graphical pair $((G,g),(H,h))$ without loops consists here of graphs $G$ and $H$…

Let $G$ be a simple graph, and let $k$ be a positive integer. Suppose that the vertices of $G$ whose degrees are divisible by $k$ form an induced forest. An equitable factorization…

1-factorization conjecture. Every regular graph of sufficiently high degree has a 1-factorization.

Plantholt–Tipnis multigraph 1-factorization conjecture. If the degree of $G$ is at least $rn$, then $G$ is 1-factorizable.

Chetwynd–Hilton 1-factorization conjecture. Every regular simple graph of order $2n$ and degree at least $n$ is 1-factorizable.