New i-Perfect Cycle Decompositions via Vertex Colorings of Graphs

Generalizing a result by Buratti etal.[M. Buratti, F. Rania, and F. Zuanni, Some constructions for cyclic perfect cycle systems, Discrete Math 299 (2005), 33-48], we present a construction for i-perfect k-cycle decompositions of the complete m-partite graph with parts of size k. These decompositions are sharply vertex-transitive under the additive group of ZkxR, with R a suitable ring of order m. The construction works whenever a suitable i-perfect mapf:Zk?R exists. We show that for determining the set of all triples (i,k,m) for which such a map exists, it is crucial to calculate the chromatic numbers of some auxiliary graphs. We completely determine this set except for one special case where k>1,000 is the product of two distinct primes, i>2 is even, and gcd(m,25)=5. This result allows us to obtain a plethora of new i-perfect k-cycle decompositions of the complete graph of order vk (mod 2k) with k odd. In particular, if k is a prime, such a decomposition exists for any possible i provided that gcd(v/k>,9)3.