Bounded-Degree Planar Graphs Do Not Have Bounded-Degree Product Structure

Product structure theorems are a collection of recent results that have been used to resolve a number of longstanding open problems on planar graphs and related graph classes. One particularly useful version states that every planar graph G is contained in the strong product of a 3-tree H, a path P, and a 3-cycle K3; written as G subset of H (R) P (R) K3. A number of researchers have asked if this theorem can be strengthened so that the maximum degree in H can be bounded by a function of the maximum degree in G. We show that no such strengthening is possible. Specifically, we describe an infinite family g of planar graphs of maximum degree 5 such that, if an n-vertex member G of g is isomorphic to a subgraph of H (R) P (R) Kc where P is a path and H is a graph of maximum degree Delta and treewidth t, then Mathematics Subject Classifications: 05C76