, Volume 85, Issue 3, pp 563-579

C 4p -frame of complete multipartite multigraphs

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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 (g 1, h 1) and (g 2, h 2) are adjacent whenever ${g_{1}g_{2} \in E(G)}$ or g 1g 2 and ${h_{1}h_{2} \in E(H)}$ . Clearly ${K_{m} \otimes I_{n}}$ , 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 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 C 2k -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 C 4p -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 C 4p -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.

The first author thanks Jawaharlal Nehru Memorial Fund for the financial assistance through grant No. SU-A/ 007/ 2011-12/ 394 and the second author thanks the Department of Science and Technology, Government of India, New Delhi for its financial support through the Grant No. DST/INSPIRE Fellowship/2011-IF110084 and the third author thanks the Department of Science and Technology, Government of India, New Delhi for its financial support through the Grant No. DST/ SR/ S4/ MS:372/ 06.