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 graph and connecting
each new vertex with vertices 
, 
, and 
of the cycle graph (DeAlba
et al. 2009, American Institute of Mathematics). In other words, it is the
graph obtained by connecting 
diamond graphs placed around
a circle with a total of 
edges, where edges
are used to connect vertices of vertex
degree 2 in neighboring diamond graphs. The
graph might therefore even more appropriately be termed
the "diamond necklace graph." The first few necklace graphs for 
, 2, ... are illustrated above. The (degenerate) 
case gives the tetrahedral
graph 
.
With the exception of 
,
-necklace graphs are unit-distance
and matchstick graphs. In fact, the 2-necklace
graph is the minimum 3-regular matchstick
graph, illustrated above in a matchstick graph drawing.
Amazingly and amusingly, the 
-necklace graph has exactly 
independent edge sets
(E. Weisstein Feb. 10, 2026).
The 
-necklace
graph is the 
special case of a graph based on the truncated square lattice,
known in this work as the truncated square
lattice graph.
