Journal of the Australian Mathematical Society

Research Article

Dacey Graphs

David P. Sumnera1

a1 Department of Mathematics University of South Carolina Columbia, South Carolina 29208, U. S. A.

In this paper our graphs will be finite, undirected, and without loops or multiple edges. We will denote the set of vertices of a graph G by V(G). If G is a graph and u, vxs2208V(G), then we will write u xs223C v to denote that u and v are adjacent and u ≁ v otherwise. If A xs2286 V(G), then we let N(A) = {uxs2208 V(G)|u xs223C a for each a xs2208A}. However we write N(v) instead of N({v}). When there is no chance of confusion, we will not distinguish between a subset A xs2286 V(G) of vertices of G and the subgraph that it induces. We will denote the cardinality of a set A by |A|. The degree of a vertex v is δ(v) = |N(v)|. Any undefined terminology in this paper will generally conform with Behzad and Chartrand [1].

(Received March 13 1973)

(Revised August 09 1973)