On the structure of dense triangle-free graphs

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 delta greater than or equal to 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 delta > n/3 is either homomorphic with Cs or contains the Mobius ladder. A major tool is the observation that every triangle-free graph with delta greater than or equal to n/3 has a unique maximal triangle-free supergraph.