Chang, YanxunDukes, Peter J.Feng, Tao2021-03-012021-03-0120192019Chang, Y., Dukes, P. J., & Feng, T. (2019). Leaves for packings with block size four. arXiv. https://arxiv.org/abs/1905.12151https://arxiv.org/abs/1905.12151http://hdl.handle.net/1828/12746We consider maximum packings of edge-disjoint 4-cliques in the complete graph Kn. When n 1 or 4 (mod 12), these are simply block designs. In other congruence classes, there are necessarily uncovered edges; we examine the possible ‘leave’ graphs induced by those edges. We give particular emphasis to the case n 0 or 3 (mod 12), when the leave is 2-regular. Colbourn and Ling settled the case of Hamiltonian leaves in this case. We extend their construction and use several additional direct and recursive constructions to realize a variety of 2-regular leaves. For various subsets S {3, 4, 5, . . . }, we establish explicit lower bounds on n to guarantee the existence of maximum packings with any possible leave whose cycle lengths belong to S.enPreprintDepartment of Mathematics and Statistics