# regular

## Friendly partitions ★★

Author(s): DeVos

A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.

Problem   Is it true that for every , all but finitely many -regular graphs have friendly partitions?

Keywords: edge-cut; partition; regular

## Nearly spanning regular subgraphs ★★★

Author(s): Alon; Mubayi

Conjecture   For every and every positive integer , there exists so that every simple -regular graph with has a -regular subgraph with .

Keywords: regular; subgraph

## r-regular graphs are not uniquely hamiltonian. ★★★

Author(s): Sheehan

Conjecture   If is a finite -regular graph, where , then is not uniquely hamiltonian.

Keywords: hamiltonian; regular; uniquely hamiltonian