Cycle Frames of Complete Multipartite Multigraphs ‐ III

For two graphs G and H their wreath productG ⊗ H has vertex set V ( G ) × V ( H ) in which two vertices ( g 1 , h 1 ) and ( g 2 , h 2 ) are adjacent wheneverg 1 g 2 ∈ E ( G )org 1 = g 2andh 1 h 2 ∈ E ( H ) . Clearly,K m ⊗ I n , where I n is an independent set on n vertices, is isomorphic to the complete m ‐partite graph in which each partite set has exactly n vertices. A 2‐regular subgraph of the complete multipartite graphK m ⊗ I ncontaining vertices of all but one partite set is called partial 2 ‐factor . For an integer λ, G ( λ ) denotes a graph G with uniform edge multiplicity λ. Let J be a set of integers. If( K m ⊗ I n ) ( λ )can be partitioned into edge‐disjoint partial 2‐factors consisting cycles of lengths from J , then we say that( K m ⊗ I n ) ( λ )has a( J , λ ) ‐cycle frame . In this paper, we show that for m ≥ 3 and k ≥ 2 , there exists a ( 2 k , λ ) ‐cycle frame of( K m ⊗ I n ) ( λ )if and only if ( m − 1 ) n ≡ 0 ( mod 2 k ) and λ n ≡ 0 ( mod 2 ) . In fact our results completely solve the existence of a ( 2 k , λ ) ‐cycle frame of( K m ⊗ I n ) ( λ ) .