As a consequence of an early result of Pach we show that every maximal triangle-free
graph is either homomorphic with a member of a specific infinite sequence of graphs or
contains the Petersen graph minus one vertex as a subgraph. From this result and further
structural observations we derive that, if a (not necessarily maximal) triangle-free graph
of order n has minimum degree δ[ges ]n/3,
then the graph is either homomorphic with a
member of the indicated family or contains the Petersen graph with one edge contracted.
As a corollary we get a recent result due to Chen, Jin and Koh. Finally, we show that every
triangle-free graph with δ>n/3 is either homomorphic with
C5 or contains the Möbius
ladder. A major tool is the observation that every triangle-free graph with
δ[ges ]n/3 has a unique maximal triangle-free supergraph.