On the number of congruence classes of paths

TL;DR

This paper determines the number of congruence classes of paths induced by graph homomorphisms, solving an open problem by Michels and Knauer, using a new formula for counting path homomorphisms.

Contribution

It provides a new formula for counting homomorphisms between paths and applies it to settle an open problem on congruence classes.

Findings

Exact count of congruence classes for paths

New formula for path homomorphisms

Solved an open problem in graph theory

Abstract

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.