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Triangular Snake Graph

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TriangularSnakes

The triangular snake graph TS_n is the graph on n vertices with n odd defined by starting with the path graph P_n and adding edges (2i-1,2i+1) for i=1, ..., (n-1)/2. The first few are illustrated above, and special cases are summarized in the following table.

nTS_n
1singleton graph K_1
3triangle graph C_3
5butterfly graph

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 2025), which is equivalent to when n=1,3 (mod 8). Triangular snakes are also geodetic.

See also

Butterfly Graph, Path Graph, Polyiamond, Snake Graph, Triangle Graph

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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 mK_1." Path and Cycle. Int. J. Comp. Appl. 44, 20-22, 2012.

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Triangular Snake Graph

Cite this as:

Weisstein, Eric W. "Triangular Snake Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TriangularSnakeGraph.html

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