5 problems

Bollobás–Scott's conjecture. Every graph $G$ has a bisection in which every vertex $v$ has at least

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

Triangle-free subcubic bisection conjecture. Every weighted triangle-free subcubic graph other than $K_{1,3}$ 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.

Let $G$ be a graph, and let a bisection $H$ be a bipartite spanning subgraph of $G$ whose two partition sets differ in size by at most one. For $v\in V(G)$, let $d_G(v)$ and…