Packing Trees into 1planar Graphs We introduce and study the 1planar packing problem: Given k graphs with n vertices (G_1, dots , G_k), find a 1planar graph that contains the given graphs as edgedisjoint spanning subgraphs. We mainly focus on the case when each (G_i) is a tree and (k3). We prove that a triple consisting of three caterpillars or of two caterpillars and a path may not admit a 1planar packing, while two paths and a special type of caterpillar always have one. We then study 1planar packings with few crossings and prove that three paths (resp. cycles) admit a 1planar packing with at most seven (resp. fourteen) crossings. We finally show that a quadruple consisting of three paths and a perfect matching with (n ge 12) vertices admits a 1planar packing, while such a packing does not exist if (n le 10).

The EdmondsGallai Decomposition for the kPiece Packing Problem Generalizing Kanekou0027s long path packing problem, Hartvigsen, Hell and Szabo consider a new type of undirected graph packing problem, called the  kpiece packing problem . A kpiece is a simple, connected graph with highest degree exactly k so in the case k1 we get the classical matching problem. They give a polynomial algorithm, a Tuttetype characterization and a Bergetype minimax formula for the kpiece packing problem. However, they leave open the question of an EdmondsGallai type decomposition. This paper fills this gap by describing such a decomposition. We also prove that the vertex sets coverable by kpiece packings have a certain matroidal structure.