Packing \(5\)-cycles into balanced complete \(n\)-partite graphs for even \(n\): the \(1\)-factor leave

Abstract

Let \( K_{g_1,g_2,\dots,g_n} \) be a complete \( n \)-partite graph with partite sets of sizes \( g_i \) for \( 1 \leq i \leq n \). A complete \( n \)-partite is balanced if each partite set has \( g \) vertices. In this paper, we will solve the problem of finding a maximum packing of the balanced complete \( n \)-partite graph, \( n \) even, with edge-disjoint 5-cycles when the leave is a 1-factor.