An induced subdivision of a graph 
in a graph 
is an induced subgraph
of 
isomorphic to a graph
subdivision of 
.
Each graph edge of 
is replaced by a graph path
of length at least one. The replacing graph paths have
pairwise disjoint interiors containing none of the original vertices,
and there are no additional edges between vertices
of the resulting subdivision (Lévêque et al. 2012).
A graph with no induced subdivision of the tetrahedral graph 
is called ISK4-free. Lévêque et al. (2012) conjectured that every
such graph has chromatic
number at most 4. Cui and Lan (2026) proved this bound for graphs
that also have no induced subgraph isomorphic
to the diamond graph 
. The bound 4 is sharp within this restricted class (Cui
and Lan 2026).
-Free Graphs." 23 Oct 2026. 
."
J. Combin. Th., Ser. B 102, 924-947, 2012.