If G is the square of a graph H,
then each vertex has a closed neighborhood which generates a complete subgraph of
G and G is the union of these complete subgraphs. Although the converse fails, it
does suggest a classification which yields a theory extensive enough to be of
independent interest. This paper develops some basic properties of what
will be called locally complete graphs. In §3 the theory is applied to the
problem of square roots, and an existence theorem is proved from which
Mukhopadhyay’s theorem [3] follows as a corollary. Based on the more general
theorem, a technique for square root determination is illustrated in the final
section.
