Internal Partitions of Regular Graphs

An internal partition of an n-vertex graph (Formula presented.) is a partition of V such that every vertex has at least as many neighbors in its own part as in the other part. It has been conjectured that every d-regular graph with (Formula presented.) vertices has an internal partition. Here we prove this for (Formula presented.). The case (Formula presented.) is of particular interest and leads to interesting new open problems on cubic graphs. We also provide new lower bounds on (Formula presented.) and find new families of graphs with no internal partitions. Weighted versions of these problems are considered as well.