Aequationes mathematicae

, Volume 85, Issue 3, pp 563–579

# C4p-frame of complete multipartite multigraphs

Article

## Abstract

For two graphs G and H their wreath product$${G \otimes H}$$ has the vertex set $${V(G) \times V(H)}$$ in which two vertices (g1, h1) and (g2, h2) are adjacent whenever $${g_{1}g_{2} \in E(G)}$$ or g1g2 and $${h_{1}h_{2} \in E(H)}$$ . Clearly $${K_{m} \otimes I_{n}}$$ , where In 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 subgraph of the complete multipartite graph $${K_m \otimes I_n}$$ containing vertices of all but one partite set is called partial factor. An H-frame of $${K_m \otimes I_n}$$ is a decomposition of $${K_m \otimes I_n}$$ into partial factors such that each component of it is isomorphic to H. In this paper, we investigate C2k-frames of $${(K_m \otimes I_n)(\lambda)}$$ , and give some necessary or sufficient conditions for such a frame to exist. In particular, we give a complete solution for the existence of a C4p-frame of $${(K_m \otimes I_n)(\lambda)}$$ , where p is a prime, as follows: For an integer m ≥  3 and a prime p, there exists a C4p-frame of $${(K_m \otimes I_n)(\lambda)}$$ if and only if $${(m-1)n \equiv 0 ({\rm {mod}} {4p})}$$ and at least one of m, n must be even, when λ is odd.

### Mathematics Subject Classification (2010)

05C70 05C51 05B30

### Keywords

Decomposition factorization frame

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