题目

英语翻译
ON THE NUMBER OF CONGRUENCE CLASSES OF PATHS
ZHICONG LIN AND JIANG ZENG
Abstract.Let Pn denote the undirected path of length n − 1.The cardinality of the set of congruence classes induced by the graph homomorphisms from Pn onto Pk 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.
Keywords:Graph,graph endomorphisms,graph homomorphisms,paths,lattice paths
1.Introduction
We use standard notations and terminology of graph theory in [3] or [6,Appendix].The graphs considered here are finite and undirected without multiple edges and loops.Given a graph G,we write V (G) for the vertex set and E(G) for the edge set.A homomorphism from a graph G to a graph H is a mapping f :V (G) → V (H) such that the images of adjacent vertices are adjacent.An endomorphism of a graph is a homomorphism from the graph to itself.Denote b

提问时间：2020-06-13

答案

同余类的路径ZHICONG林,江曾摘要的数量.令Pn表示无向路径长度为n - 1.确定从的Pn到PK的图形同态诱导的同余类的集合的基数.这解决的一个公开问题的的米歇尔斯和克瑙尔（Disc.数学系,309（2009）5352-5359）.我们的结...