TOPICS
Triangular Snake Graph
The triangular snake graph 
is the graph on 
vertices with 
odd defined by starting with the path
graph 
and adding edges 
for 
,
..., 
.
The first few are illustrated above, and special cases are summarized in the following
table.
Triangular snakes are unit-distance and matchstick by construction, perfect.
They are graceful when the number of triangles
is congruent to 0 or 1 (mod 4) (Moulton 1989, Gallian 2018), which is equivalent
to when 
.
Triangular snakes are also geodetic.
See also
Butterfly Graph, Path Graph, Polyiamond, Triangle GraphExplore with Wolfram|Alpha
References
Clancy, K.; Haythorpe, M.; and Newcombe, A. §4.5.1 in "A Survey of Graphs with Known or Bounded Crossing Numbers." 15 Feb 2019, pp. 58-59. https://arxiv.org/abs/1901.05155.Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin., Dynamic Survey DS6, Oct. 30, 2025. https://doi.org/10.37236/27.House of Graphs. Triangular Snake Graphs. Singleton Graph, Triangle K3, Butterfly Graph, and Triangular Snake Graph 7.Moulton, D. "Graceful Labelings of Triangular Snakes." Ars Combin. 28, 3-13, 1989.Rajan, B.; Rajasingh. I.; and Vasanthi Beulah, P. "Crossing Number of Join of Triangular Snake with
." Path and Cycle. Int. J. Comp. Appl. 44,
20-22, 2012.Referenced on Wolfram|Alpha
Triangular Snake GraphCite this as:
Weisstein, Eric W. "Triangular Snake Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TriangularSnakeGraph.html