The Eulerian Cycle Decomposition Conjecture, by Chartrand, Jordon and Zhang, states that if the minimum number of odd cycles in a cycle decomposition of an Eulerian graph of size is the maximum number of odd cycles in such a cycle decomposition is and is an integer such that where and are of the same parity, then there is a cycle decomposition of with exactly odd cycles. This conjecture is verified for the line graph of the complete graph.