P ⃗_2k-Factorization of Complete Bipartite Symmetric Multi-digraph

Abstract

The graph  be the path graph on  vertices with  edges and  be the directed path on  vertices with  arcs. The graph  be the complete bipartite graph with two partite sets  and  having  and  elements respectively and the graph  be the complete bipartite symmetric multi-digraph. In this paper we will discuss and established the conditions for the existence of the path decomposition of factorization of complete bipartite symmetric multi-digraph. The following necessary and sufficient conditions for the existence of the path factorization of graphs are as follows, if  and  then the graph  has  factorization, where  and  and  are positive integers.