Equipartite and almost-equipartite gregarious 4-cycle systems

• Published in: Discrete mathematics Vol. 308; no. 5; pp. 696 - 714
• Main Authors: Billington, Elizabeth J., Hoffman, D.G.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Kidlington Elsevier B.V 28.03.2008; Elsevier
• Subjects: 4-cycle; Algebra; Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Gregarious cycle; Mathematics; Multipartite graph decomposition; Sciences and techniques of general use; 4-cycle; Multipartite graph decomposition; Gregarious cycle; Graph decomposition; Multipartite graph; Complete graph; Decomposition method; Discrete mathematics; Combinatorics
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2007.07.056

A 4-cycle decomposition of a complete multipartite graph is said to be gregarious if each 4-cycle in the decomposition has its vertices in four different partite sets. Here we exhibit gregarious 4-cycle decompositions of the complete equipartite graph K n ( m ) (with n ⩾ 4 parts of size m) whenever a 4-cycle decomposition (gregarious or not) is possible, and also of a complete multipartite graph in which all parts but one have the same size. The latter complete multipartite graph, K n ( m ) , t , having n parts of size m and one part of size t, has a gregarious 4-cycle decomposition if and only if (i) n ⩾ 3 , (ii) t ⩽ ⌊ m ( n - 1 ) / 2 ⌋ and (iii) a 4-cycle decomposition exists, that is, either m and t are even or else m and t are both odd and n ≡ 0 ( mod 8 ) .