5 research outputs found
On the number of congruence classes of paths
Let denote the undirected path of length . The cardinality of the
set of congruence classes induced by the graph homomorphisms from onto
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.Comment: 11 pages, 2 figures, to appear in Discrete Mathematic
On monoids of endomorphisms of a cycle graph
In this paper we consider endomorphisms of an undirected cycle graph from
Semigroup Theory perspective. Our main aim is to present a process to determine
sets of generators with minimal cardinality for the monoids and
of all weak endomorphisms and all endomorphisms of an undirected
cycle graph with vertices. We also describe Green's relations and
regularity of these monoids and calculate their cardinalities