Internal Partitions of Regular Graphs

Abstract

AbstractAn internal partition of an n‐vertex graph 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 vertices has an internal partition. Here we prove this for . The case is of particular interest and leads to interesting new open problems on cubic graphs. We also provide new lower bounds on and find new families of graphs with no internal partitions. Weighted versions of these problems are considered as well.