Originator: Saeeid Akbari and Mikio Kano (presented by Sogol Jahanbekan - REGS 2011)
Definitions: An $\{a,b\}$-factor of a graph $G$ is a spanning subgraph of $G$, say $G'$, such that $d_{G'}(v)\in \{a,b\}$ for every $a,b\in V(G)$.
Background: It is known that every $5$-regular graph $G$ decomposes into two $\{2,3\}$-factors. In fact, if $|E(G)|$ is even, then the two subgraphs can also be required to have the same degree list (see [COW]).
A similar question is motivated by a version of Tutte's 3-flow Conjecture: Every $4$-edge connected $5$-regular graph has an edge orientation in which every outdegree is $4$ or $1$ (see [AP]).
Conjecture: (Akbari-Kano, 2011) Every $5$-regular graph can be decomposed into two $\{1,4\}$-factors.
Comments: Later in REGS 2011, Sogol Jahabekan proved the conjecture using the Combinatorial Nullstellensatz. It turns out that, appropriately generalized, the technique of proof gives a uniform simple proof of the original conjecture of Akbari and Kano, which is that when $k\ge3$ every $k$-regular graph has an edge-labeling using labels $1$ and $2$ such that at each vertex the sum of the labels is divisible by $3$.
References:
[AP]
Noga Alon and Powel Pralat; Modular orientations of random and quasi-random
regular graphs.
[COW]
Jeong Ok Choi, Lale Ozkahya, and Douglas B. West;
Degree-splittability of multigraphs and caterpillars,
Proc. 41st SE Conf., Congressus Numer. 202 (2010), 137-147.