A triangle-replaced graph is a cubic graph in which each vertex is replaced by a triangle graph such that each vertex of the triangle is connected to one of the originally adjacent vertices of a graph .
Bryant and Dean (2014) consider a generalization to a -replaced graph, in which the vertices of a -regular graph are replaced by copies of the complete graph . Such graphs provide counterexamples to the conjecture that there are a finite number of connected vertex-transitive graphs that have no Hamilton decomposition. The smallest counterexample is given by the -replaced graph obtained from the multigraph obtained from the cubical graph by doubling its edges.
Special cases of triangle-replaced graphs are summarized in the following table.