Characterizations of postman sets

Using results by McKee and Woodall on binary matroids, we show that the set of postman sets has odd cardinality, generalizing a result by Toida on the cardinality of cycles in Eulerian graphs. We study the relationship between TT-joins and blocks of the underlying graph, obtaining a decomposition of postman sets in terms of blocks. We conclude by giving several characterizations of TT-joins which are postman sets.