Vol. 47, No. 1, 1973

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Locally complete graphs

Henry S. Sharp, Jr.

Vol. 47 (1973), No. 1, 243–250

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.

Mathematical Subject Classification 2000
Primary: 05C99
Received: 31 January 1972
Published: 1 July 1973
Henry S. Sharp, Jr.