The Planar Tree Packing Theorem

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of Garc\'ia et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.Comment: Full version of our SoCG 2016 pape

Asymptotically optimal covering designs

A (v,k,t) covering design, or covering, is a family of k-subsets, called blocks, chosen from a v-set, such that each t-subset is contained in at least one of the blocks. The number of blocks is the covering's size}, and the minimum size of such a covering is denoted by C(v,k,t). It is easy to see that a covering must contain at least (v choose t)/(k choose t) blocks, and in 1985 R\"odl [European J. Combin. 5 (1985), 69-78] proved a long-standing conjecture of Erd\H{o}s and Hanani [Publ. Math. Debrecen 10 (1963), 10-13] that for fixed k and t, coverings of size (v choose t)/(k choose t) (1+o(1)) exist (as v \to \infty). An earlier paper by the first three authors [J. Combin. Des. 3 (1995), 269-284] gave new methods for constructing good coverings, and gave tables of upper bounds on C(v,k,t) for small v, k, and t. The present paper shows that two of those constructions are asymptotically optimal: For fixed k and t, the size of the coverings constructed matches R\"odl's bound. The paper also makes the o(1) error bound explicit, and gives some evidence for a much stronger bound