# 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?

## Nearly spanning regular subgraphs ★★★

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

## 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