Internal Partitions of Regular Graphs

AbstractAn internal partition of an n-vertex graph G = (V,E) is a partition of Vsuch that every vertex has at least as many neighbors in its own part as in theother part. It has been conjectured that every d-regular graph with n > N(d)vertices has an internal partition. Here we prove this for d = 6. The cased = n−4 is of particular interest and leads to interesting new open problems oncubic graphs. We also provide new lower bounds on N(d) and find new familiesof graphs with no internal partitions. Weighted versions of these problems areconsidered as well.1. IntroductionIt is well-known that every finite graph G = (V,E) has an external partition,i.e., a splitting of V into two parts such that each vertex has at least half ofits neighbors in the other part. This is, e.g., true for G’s max-cut partition.Much less is known about the internal partition problem in which V is split intotwo non-empty parts, such that each vertex has at least half of its neighbors inits own part. Not all graphs have an internal partition and their existence isproved only for certain classes of graphs. Several investigators have raised theconjecture that for every d there is an n