
The author investigates sufficient conditions for the existence of a 5-cycle trade of volume \(t\) and foundation \(v.\) This interesting survey includes recent results on volumes of 5-cycle trades. It is a continuation of articles by \textit{A. Rosa} [Cas. Pestovni Mat. 91, 53-62 (1966; Zbl 0151.33501)], by \textit{A. Rosa} and \textit{S. Znám} [Discrete Math. 128, No. 1-3, 305-316 (1994; Zbl 0797.05063)] and by \textit{D. E. Bryant} [Australas. J. Comb. 15, 161-176 (1997; Zbl 0880.05017)]. A 5-cycle trade of volume \(t\) is a graph \(G\) whose edge set can be partitioned into \(t\) 5-cycles in at least two ways, such that the two collections of 5-cycles have no 5-cycles in common. The foundation \(v\) of a trade is the number of distinct vertices occurring in the graph \(G.\) Let \(E(v)\) be the set of all \(t\) such that there exists a 5-cycle trade of volume \(t\) with foundation \(v\) and let \(P(v)\) be the set of the possible volumes for a 5-cycle trade on \(v\) vertices. The author proves that, for all non-negative integers \(v\), \(E(v) = P(v).\) Tables of upper and lower bounds on the volume of a 5-cycle trade on \(v\) vertices and large, small and middle trade volumes are given.
230100 Mathematics, foundation of a trade, volume of a trade, C1, Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.), 5-cycle, 5-cycle trade, Paths and cycles, Mathematics
230100 Mathematics, foundation of a trade, volume of a trade, C1, Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.), 5-cycle, 5-cycle trade, Paths and cycles, Mathematics
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