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A DIFFERENCE SET METHOD FOR CIRCULANT DECOMPOSITIONS OF COMPLETE PARTITE GRAPHS INTO GREGARIOUS 4-CYCLES

  • Kim, Eun-Kyung (Department of Mathematics Pusan National University) ;
  • Cho, Young-Min (Department of Mathematics Pusan National University) ;
  • Cho, Jung-Rae (Department of Mathematics Pusan National University)
  • Received : 2010.06.23
  • Accepted : 2010.11.10
  • Published : 2010.11.30

Abstract

The complete multipartite graph $K_{n(m)}$ with n $ {\geq}$ 4 partite sets of size m is shown to have a decomposition into 4-cycles in such a way that vertices of each cycle belong to distinct partite sets of $K_{n(m)}$, if 4 divides the number of edges. Such cycles are called gregarious, and were introduced by Billington and Hoffman ([2]) and redefined in [3]. We independently came up with the result of [3] by using a difference set method, and improved the result so that the composition is circulant, in the sense that it is invariant under the cyclic permutation of partite sets. The composition is then used to construct gregarious 4-cycle decompositions when one partite set of the graph has different cardinality than that of others. Some results on joins of decomposable complete multipartite graphs are also presented.

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Cited by

  1. CIRCULANT DECOMPOSITIONS OF CERTAIN MULTIPARTITE GRAPHS INTO GREGARIOUS CYCLES OF A GIVEN LENGTH vol.30, pp.3, 2010, https://doi.org/10.7858/eamj.2014.021
  2. A REMARK ON CIRCULANT DECOMPOSITIONS OF COMPLETE MULTIPARTITE GRAPHS BY GREGARIOUS CYCLES vol.33, pp.1, 2010, https://doi.org/10.7858/eamj.2017.007