The 1-factorization conjecture for regular graphs

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The 1-factorization conjecture for regular graphs

   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 and $N\geq n$, or if $n$ is even and $N\geq n-1$, then $G$ has a 1-factorization. This conjecture gives degree thresholds guaranteeing 1-factorizations beyond the bipartite case; the source presents it as a related general conjecture, without stating a resolution.

   source: Robert W. Donley, S. James Gates, Tristan Hübsch and Rishi Nath, “A combinatorial introduction to Adinkras”, arXiv:2410.12834 (2024).