par Dujmović, Vida V.;Joret, Gwenaël 
;Micek, Piotr;Morin, Pat P.;Wood, David
Référence The electronic journal of combinatorics, 31, 2, P2.51
Publication Publié, 2024
;Micek, Piotr;Morin, Pat P.;Wood, David Référence The electronic journal of combinatorics, 31, 2, P2.51
Publication Publié, 2024
Article révisé par les pairs
| Résumé : | 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 ⊆ H ⊠ P ⊠ 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 ⊠ P ⊠ Kc where P is a path and H is a graph of maximum degree ∆ and treewidth t, then t∆c ≥ 2Ω(√log log n). |
