Resolvable 4-cycle group divisible designs with two associate classes: Part size even

• Published in: Discrete mathematics Vol. 308; no. 2; pp. 303 - 307
• Main Authors: Billington, Elizabeth J., Rodger, C.A.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier B.V 06.02.2008; Elsevier
• Subjects: 2-factorizations; Applied sciences; Associate classes; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Cycle systems; Designs and configurations; Exact sciences and technology; Information retrieval. Graph; Mathematics; Resolvable; Sciences and techniques of general use; Theoretical computing; Resolvable; Associate classes; Cycle systems; 2-factorizations; Design; Graph decomposition; Necessary and sufficient condition; Decomposition method; Factorization; Cycle(graph)
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2006.11.043

Let λ 1 K a denote the graph on a vertices with λ 1 edges between every pair of vertices. Take p copies of this graph λ 1 K a , and join each pair of vertices in different copies with λ 2 edges. The resulting graph is denoted by K ( a , p ; λ 1 , λ 2 ) , a graph that was of particular interest to Bose and Shimamoto in their study of group divisible designs with two associate classes. The existence of z-cycle decompositions of this graph have been found when z ∈ { 3 , 4 } . In this paper we consider resolvable decompositions, finding necessary and sufficient conditions for a 4-cycle factorization of K ( a , p ; λ 1 , λ 2 ) (when λ 1 is even) or of K ( a , p ; λ 1 , λ 2 ) minus a 1-factor (when λ 1 is odd) whenever a is even.