As defined by DeAlba et al. (2009) and references on the American Institute of Mathematics Minimum Rank Graph Catalogs, an 
-necklace graph is a cubic graph
on 
vertices obtained from a cycle
graph 
by adding an additional vertex associated with every third vertex 
of the cycle and connecting each new vertex with vertices
, 
, and 
of the cycle. The first few necklace graphs for 
, 2, ... are illustrated above. The (degenerate) 
case gives the tetrahedral
graph 
.
Necklace graphs are unit-distance and matchstick graphs.
