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.

1	singleton graph
3	triangle graph
5	butterfly 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 2018), which is equivalent to when . Triangular snakes are also geodetic.

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. DS6. Dec. 21, 2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.

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.