5 problems

Kempe-equivalence conjecture. The graph $G$ is perfectly contractile if and only if, for every replication graph $H$ of an arbitrary induced subgraph of $G$ and every…

Degree-swappability conjecture. Every $3$-connected graph distinct from the complete graph and the triangular prism is degree-swappable.

Let $G$ be a simple graph. Two $k$-edge-colourings are Kempe-equivalent if one can be reached from the other through a series of Kempe changes using colours from $\{1,\ldots,k\}$.…

For a graph $G$, a $t$-colouring is a proper colouring using $t$ colours, and two colourings are Kempe equivalent when they are connected by a sequence of Kempe changes, each swapp…

Let a quintic graph be a 5-regular graph, and let two 1-factorisations be Kempe equivalent if one can be obtained from the other by a sequence of Kempe switches. Kempe-class conjec…