Path and cycle decompositions of complete equipartite graphs: 3 and 5 parts

• Published in: Discrete mathematics Vol. 310; no. 2; pp. 241 - 254
• Main Authors: Billington, Elizabeth J., Cavenagh, Nicholas J., Smith, Benjamin R.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Kidlington Elsevier B.V 28.01.2010; Elsevier
• Subjects: Algebra; Combinatorics; Combinatorics. Ordered structures; Complete equipartite graph; Cycle decomposition; Exact sciences and technology; Mathematics; Path decomposition; Sciences and techniques of general use; Cycle decomposition; Complete equipartite graph; Path decomposition; Graph decomposition; Necessary and sufficient condition; Complete graph; Decomposition method; Discrete mathematics; Combinatorics; Cycle(graph)
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2008.09.003

In 1998 Cavenagh [N.J. Cavenagh, Decompositions of complete tripartite graphs into k -cycles, Australas. J. Combin. 18 (1998) 193–200] gave necessary and sufficient conditions for the existence of an edge-disjoint decomposition of a complete equipartite graph with three parts, into cycles of some fixed length k . Here we extend this to paths, and show that such a complete equipartite graph with three partite sets of size m , has an edge-disjoint decomposition into paths of length k if and only if k divides 3 m 2 and k < 3 m . Further, extending to five partite sets, we show that a complete equipartite graph with five partite sets of size m has an edge-disjoint decomposition into cycles (and also into paths) of length k with k ⩾ 3 if and only if k divides 10 m 2 and k ⩽ 5 m for cycles (or k < 5 m for paths).