4 problems

Triangle-free subcubic bisection conjecture. Every weighted triangle-free subcubic graph other than $K_{1,3}$ has a bisection of weight at least

Bounded-degree bisection conjecture. If $k$ is odd, then $G$ has a bisection of weight at least

Ando's colouring conjecture. A cubic graph $G$ admits a $2$-vertex colouring $c_V$ such that the monochromatic induced subgraphs are isomorphic.

Ando's conjecture. Every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic.