Lambda-fold 2-perfect 6-cycle systems in equipartite graphs

• Published in: Discrete mathematics Vol. 311; no. 21; pp. 2423 - 2427
• Main Authors: Billington, Elizabeth J., Hoffman, D.G.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 06.11.2011; Elsevier
• Subjects: 2-perfect cycle system; 6-cycle system; Algebra; Applied sciences; Combinatorics; Combinatorics. Ordered structures; Complete equipartite graph; Computer science; control theory; systems; Decomposition; Exact sciences and technology; Graphs; Information retrieval. Graph; Joining; Mathematical analysis; Mathematics; Sciences and techniques of general use; Theoretical computing; 2-perfect cycle system; 6-cycle system; Complete equipartite graph; Edge(graph); Necessary and sufficient condition; Complete graph; Distance graph; Decomposition method; Discrete mathematics; Combinatorics; Cycle(graph); Cycle time
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2011.06.026

A 6-cycle system of a graph G is an edge-disjoint decomposition of G into 6-cycles. Graphs G, for which necessary and sufficient conditions for existence of a 6-cycle system have been found, include complete graphs and complete equipartite graphs. A 6-cycle system of G is said to be 2-perfect if the graph formed by joining all vertices distance 2 apart in the 6-cycles is again an edge-disjoint decomposition of G, this time into 3-cycles, since the distance 2 graph in any 6-cycle is a pair of disjoint 3-cycles. Necessary and sufficient conditions for existence of 2-perfect 6-cycle systems of both complete graphs and complete equipartite graphs are known, and also of λ-fold complete graphs. In this paper, we complete the problem, giving necessary and sufficient conditions for existence of λ-fold 2-perfect 6-cycle systems of complete equipartite graphs. ► We consider lambda-fold 2-perfect 6-cycle systems of complete equipartite graphs. ► We give necessary and sufficient conditions for existence of these. ► This completely solves the existence problem for all lambda.