On the number of congruence classes of paths
  Let $P_n$ denote the undirected path of length $n-1$. The cardinality of the set of congruence classes induced by the graph homomorphisms from $P_n$ onto $P_k$ is determined. This settles an open problem of   Michels and Knauer (Disc. Math., 309\ (2009)\ 5352-5359). Our result is based on a new proven formula of the number of homomorphisms between paths. 
Zhicong Lin, Jiang Zeng
math.CO