The Sierpiński gasket graph of order 
is the graph obtained from the connectivity
of the Sierpiński sieve. The first few Sierpiński
gasket graphs are illustrated above.
is also known as the Hajós graph or the 2-sun graph (Brandstädt et al. 1987, p. 18).
Its analog in three dimensions may be termed the Sierpiński tetrahedron graph, and it can be further generalized to higher dimensions (D. Knuth, pers. comm., May 1, 2022).
Teguia and Godbole (2006) investigated the properties of the Sierpiński gasket graphs and proved that they are Hamiltonian and pancyclic.
The graph 
has 
vertices (OEIS
A067771) and 
edges (OEIS A000244;
Teguia and Godbole 2006). It has graph diameter 
and domination
number 1 for 
,
2 for 
,
and 
for 
(Teguia and Godbole 2006).
The Sierpiński gasket graphs are uniquely 3-colorable up to permutation of colors (D. Knuth, pers. comm., Apr. 11,
2022), meaning the number of distinct colorings of 
is 
for all 
, as illustrated above for 
and 
. Interestingly, unique coloring
follows directly from the symmetry of these graphs without
needing to explicitly swap colors. The generalizations of these graphs
to higher odd dimensions are also uniquely colorable (D. Knuth, pers. comm.,
May 1, 2022).
Sierpiński gasket graphs are implemented in the Wolfram Language as GraphData[
"SierpinskiGasket",
n
].
