SERIE A | MATHEMATIK 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 speciic innnite sequence of 3-colourable graphs or it 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 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 > n=3 is either homomorphic with C5 or it contains the MM obius ladder as a subgraph. A major tool is the observation that every triangle-free graph with n=3 has a unique maximal triangle-free supergraph.