On Eulerian Regular Complete \(5\)-Partite Graphs and a Cycle Decomposition Problem

An Eulerian graph \( G \) of size \( m \) is said to satisfy the Eulerian Cycle Decomposition Conjecture if the minimum number of odd cycles in a cycle decomposition of \( G \) is \( a \), the maximum number of odd cycles in a cycle decomposition is \( b \), and \( \ell \) is an integer such that \( a \leq \ell \leq b \) where \( \ell \) and \( m \) are of the same parity, then there is a cycle decomposition of \( G \) with exactly \( \ell \) odd cycles. Several regular complete \( 5 \)-partite graphs are shown to have this property.